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UNIVERSITY  OF  CALIFORNIA 
SAN  DIEGO 

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INTRODUCTION 

TO 

ANALYTICAL  MECHANICS 


•The 


THE  MACMILLAN    COMPANY 

NEW  YORK  BOSTON  CHICAGO 

DALLAS  SAN    FRANCISCO 

MACMILLAN  &  CO..  Limited 

LONDON      BOMBAY       CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  Ltd. 

TORONTO 


INTRODUCTION 

TO 

ANALYTICAL   MECHANICS 


BY 


ALEXANDER  ZIWET 

If 

PROFESSOR    OF    MATHEMATICS    IN    THE    UNIVERSITY    OF    MICHIGAN 


AND 

PETER  FIELD,  Ph.D. 

ASSISTANT    PROFESSOR    OF    MATHEMATICS    IN    THE    UNIVERSITY    OF    MICHIGAN 


Neb)  ^orlt 
THE  MACMILLAN  COMPANY 

LONDON:   MACMILLAN  &  CO.,  Ltd. 
1912 


Copyright,  1912, 
By  the   MACMILLAN   COMPANY. 


Set  up  and  electrotyped.      Published  March,  1912. 


Nortoooti  ISrrss : 
Berwick  &  Smith  Co.,  Norwood   Mass  ,  U.S  A. 


PREFACE. 

Ov)(  ws  OeXo/xev,  dAA'  a>s  SvvdfiiOa. 

The  present  volume  is  intended  as  a  brief  introduction 
to  mechanics  for  junior  and  senior  students  in  colleges  and 
universities.  It  is  based  to  a  large  extent  on  Ziwet's  Theo- 
retical Mechanics;  but  the  applications  to  engineering  are 
omitted,  and  the  analytical  treatment  has  been  broadened. 
No  knowledge  of  differential  equations  is  presupposed,  the 
treatment  of  the  occurring  equations  being  fully  explained. 
It  is  believed  that  the  book  can  readily  be  covered  in  a  three- 
hour  course  extending  throughout  a  year.  For  a  shorter 
course,  requiring  half  this  time,  the  following  selection  may 
be  made:  Chapters  1,  2,  3  (omitting  Arts.  81-95),  4  (omitting 
Arts.  114-150),  5  to  12  (omitting  Arts.  244-268),  13  and  14 
(omitting  Arts.  340-355). 

While  more  prominence  has  been  given  to  the  analytical 
side  of  the  subject,  the  more  intuitive  geometrical  ideas  are 
generally  made  to  precede  the  analysis.  In  doing  this  the 
idea  of  the  vector  is  freely  used;  but  it  has  seemed  best  to 
avoid  the  special  methods  and  notations  of  vector  analysis. 
This  has  been  done  with  re.uctance;  the  time  has  certainly 
come  for  introducing  these  methods  in  the  very  elements  of 
mechanics.     But  this  must  be  left  to  another  opportunity. 

That  many  important  subjects  had  to  be  omitted  is  another 
restriction  arising  from  the  nature  and  purpose  of  thi.s  volume. 
While  the  selection  of  topics  has  been  considered  most  care- 
fully it  can  hardly  be  expected  to  meet  everybody's  approval. 
The  aim  has  been  not  only  to  select  material  useful  to  the 
beginning  student  of  mathematics  and  physical  science,  but 


VI  PREFACE 

at  the  same  time  to  give  the  reader  a  general  view  of  the 
science  of  mechanics  as  a  whole,  a  broad  enough  foundation 
for  further  study. 

References  to  other  works  have  been  used  sparingly.  It 
seemed  hardly  necessary  to  refer  to  such  standard  works  as 
those  of  Thomson  and  Tait,  Routh,  Schell,  Appell,  Kirch- 
hoff,  etc.,  which  are  found  in  any  good  college  library.  But 
it  did  seem  desirable  to  refer  in  a  few  cases  to  works  where 
fuller  information  can  be  found  on  subjects  somewhat  out 
of  the  range  of  the  ordinary  text-book  on  mechanics.  The 
fourth  volume  of  the  Encyklopddie  der  mathematischen 
Wissenschaften,  especially  the  articles  by  P.  Stackel,  should 
be  consulted  by  the  more  advanced  student. 

Alexander  Ziwet, 
Peter  Field. 
University  of  Michigan, 
February,  1912. 


CONTENTS. 


INTRODUCTION. 


Page 
1 


PART   I:     KINEMATICS. 

CHAPTER  I:  Rectilinear  motion  of  a  point. 

1.  Velocity  and   acceleration  in  rectilinear  mo- 

tion    3 

2.  Examples  of  rectilinear  motion 10 

CHAPTER  II:   Translation  and  rotation 21 

CHAPTER  III:  Curvilinear  motion  of  a  point. 

1.  Relative  velocity;  composition  and  resolu- 

tion of  velocities 26 

2.  Velocity  in  curvilinear  motion 30 

3.  Acceleration  in  curvilinear  motion 35 

4.  Examples  of  curvilinear  motion 42 

(a)  Constant  acceleration 42 

(5)  The  pendulum .  47 

(c)  Simple  harmonic  motion 54 

(d)  Compound  harmonic  motion 59 

(e)  "Wave  motion 63 

(/)  Curvilinear  compound  harmonic  mo- 
tion    67 

(g)  Central  motion 72 

CHAPTER  IV:  Velocities  in  the  rigid  body. 

1.  Geometrical  discussion 83 

2.  Analytical  discussion 91 

3.  Plane  motion 98 

CHAPTER  V:   Accelerations  in  the  rigid  body 107 

CHAPTER  VI:   Relative  motion 117 

vii 


Vlll 


CONTENTS 


PART   II:    STATICS. 

Page 

CHAPTER  VII:  Mass;  density 120 

CHAPTER  VIII:  Moments  and  centers  of  mass 124 

CHAPTER  IX:  Momentum;  force;  energy 132 

CHAPTER  X:  Statics  of  the  particle 142 

CHAPTER  XI:  Statics  of  the  rigid  body. 

1.  Concurrent  forces 149 

2.  Parallel  forces 151 

3.  Theory  of  couples 158 

4.  Complanar  forces 165 

5.  The  general  system  of  forces 170 

6.  Constraints;  friction 178 

CHAPTER  XII:  Theory  of  attractive  forces. 

1.  Attraction 187 

2.  The  potential 196 

3.  Virtual  work 201 

PART  III:  KINETICS. 

CHAPTER  XIII:  Motion  of  a  free  particle. 

1.  The  equations  of  motion 207 

2.  Examples  of  rectilinear  motion 217 

3.  Examples  of  curvilinear  motion 229 

CHAPTER  XIV:  Constrained  motion  of  a  particle. 

1.  Introduction 248 

2.  Motion  on  a  fixed  curve 250 

3.  Motion  on  a  fixed  surface 258 

4.  The  method  of  indeterminate  multipUers  259 

5.  Lagrange's  equations  of  motion 263 

CHAPTER   XV :  The  equations  of  motion  of  a  free  rigid  body  268 

CHAPTER   XVI:  Moments  of  inertia  and  principal  axes. 

1.  Introduction 280 

2.  ElUpsoids  of  inertia 287 

3.  Distribution  of  principal  axes  in  space .  .  297 


CONTENTS  IX 

Page 

CHAPTER  XVII:  Rigid  body  with  a  fixed  axis 304 

CHAPTER  XVIII:  Rigid  body  with  a  fixed  point. 

1.  The  general  equations  of  motion 313 

2.  Motion  without  forces 320 

3.  Heavy  symmetric  top 327 

CHAPTER  XI X:  Relative  motion 335 

CHAPTER  XX:  Motion  of  a  system  of  particles. 

1.  Free  system 346 

2.  Constrained  system 348 

3.  Generalized     co-ordinates;      Lagrange's 
equations  of  motion ;  Hamilton's  principle.  352 

ANSWERS 361 

INDEX 375 


INTRODUCTION. 

1.  The  science  of  mechanics  can  be  regarded  as  an  exten- 
sion of  geometry  obtained  b}'  adjoining  tlie  ideas  of  time 
and  mass  to  tlie  idea  of  space  whicli  is  fundamental  in 
geometry.  We  are  thus  led  to  the  study  of  motion  and  of 
forces  as  the  subject-matter  of  mechanics. 

2.  By  adjoining  the  idea  of  time  alone  we  obtain  a  pre- 
liminary branch  of  mechanics,  known  as  kinematics.  It 
develops  the  ideas  of  velocity  and  acceleration  of  geometrical 
configurations  without  using  the  notion  of  mass. 

3.  The  introduction  of  mass  leads  to  numerous  new  ideas 
such  as  momentum,  force,  energy.  Owing  to  the  importance 
of  forces  in  physics  the  mechanics  of  bodies  possessing  mass 
is  often  called  dynamics.  It  may  be  divided  into  statics 
and  kinetics. 

Statics  is  the  science  of  equilibrium;  it  considers  the  con- 
ditions under  which  the  action  of  forces  produces  no  change 
of  motion.  Thus,  if  force  be  regarded  as  a  fundamental 
concept,  statics  is  independent  of  the  idea  of  time. 

Kinetics  treats  in  the  most  general  way  the  changes  of 
motion  produced  by  forces. 


PART  I:  KINEMATICS. 


CHAPTER  I. 
RECTILINEAR  MOTION  OF  A  POINT. 

1.  Velocity  and  acceleration  in  rectilinear  motion. 

4.  Consider  the  motion  of  a  point  P  along  a  fixed  straight 
line  (Fig.  1).  If  we  take  on  this  line  an  origin  0  and  a 
definite  positive  sense,  say  toward  the  right  from  0,  the 
"position  of  the  point  P  on  the  line  at  any' time  t  can  be  as- 
signed by  its  QO-ordinaie,  or  abscissa,  OP  =  s,  which  may  be 


0  P 

Fig.  1. 

any  real  number.  As  P  moves  along  the  line  its  abscissa 
■s  varies  with  the  time;  to  every  value  of  t  (at  least  within  a 
certain  interval  of  time)  corresponds  a  certain  value  of  s;  in 
other  words,  s  is  a  function  of  /.  We  assume  that  s  is  a 
continuous  function  of  t;  this  implies  that  while  P  may 
move  arbitrarily,  back  and  forth,  along  the  line,  it  does  not 
make  any  jumps,  suddenl}^  disappearing  at  one  point  and 
reappearing  at  another;  the  path  of  P  is  connected. 

5.  The  time-rate  of  change  of  the  abscissa  of  P,  i.  e.  the 
^derivative  of  s,  is  called  the  velocity  of  the  point  P;  it  is 
usually  denoted  by  the  letter  v: 

ds 
'^df 


4  laNEMATICS  [6. 

As  the  idea  of  velocity  is  fundamental  in  mechanics  it  may  be  well 
to  explain  somewhat  more  in  detail  the  genesis  of  this  idea,  the  more  so 
as  the  process  is  typical  and  recurs  frequently. 

Let  the  point  P  move  along  the  line,  or  any  segment  of  the  line, 
always  in  the  same  sense  and  so  that  equal  distances  are  always  de- 
scribed in  e'qual  times.  Such  a  motion  is  called  uniform,  and  the 
quotient  sjt  of  any  distance  OP  =  s  described,  divided  by  the  corre- 
sponding time  t,  is  called  the  velocity  of  the  uniform  motion: 


Suppose  next  that  the  point  P  does  not  move  uniformly.  The  same 
quotient,  v  =  sjt,  of  any  distance  described,  divided  by  the  time  used 
in  describing  it,  is  now  called  the  average,  or  mean,  velocity  for  that 
distance  or  time.  This  mean  velocitj^  varies  in  general  according  to  the 
distance  or  time  selected;  it  does  not  characterize  the  motion  as  a  whole. 
We  can,  however,  attach  a  definite  meaning  to  the  expression  velocity 
at  a  given  point  or  instant  if  we  define  it  as  follows. 


Hf. s 


->jAg 


P' 


Fig.  2. 


Let  s  =  OP  (Fig.  2)  be  the  abscissa  of  the  moving  point  at  the  time 
t,  s  +  As  =  OP'  its  abscissa  at  the  time  t  +  A/,  so  that  the  distance 
As  is  described  by  P  in  the  time  At;  and  let  At  be  taken  so  small  that 
P  moves  always  in  the  same  sense  as  it  describes  the  distance  As.  Then 
As/At  is  the  mean  velocity  for  the  distance  As  or  time  At.  The  Umit 
approached  by  this  quotient  as  At  approaches  zero. 

,.     As        ds 

V  =  lira-  -  =  — 

A/=o  At        dt 

is  called  the  velocity  at  the  point  P,  or  at  the  time  i. 

It  is  assumed  that  such  a  limit  exists,  i.  e.,  that  s  is  a  differentiable 
function  of  t. 

The  definition  of  velocity  as  the  time-rate  of  change  of  the  co-ordinate 

s  applies  even  in  the  case  of  uniform  motion;  for  in  this  case  we  have  as 

stated  above 

s  =  vt, 


5.]  RECTILINEAR   MOTION   OF  A   POINT  5 

where  ii  is  a  constant,  i.  e.  independent  of  t;  hence 

(Is 

dt  =  "■ 

In  non-uniform,  or  variable,  motion  the  velocity  v  varies  from  point 
to  point  and  from  time  to  time;  it  can  be  regarded  as  a  function  of  the 
distance  s  or  of  the  time  t. 

It  should  be  observed  that  in  this  whole  discussion  of  velocity  it  is 
not  essential  that  the  path  be  rectilinear,  this  assumption  being  made 
only  for  the  sake  of  simplicity.  The  discussion  applies  without  change 
when  the  point  P  describes  a  curve;  the  co-ordinate  s  then  means  the 
arc  of  the  cm've  measured  along  the  curve  from  some  origin  0  on  the 
curve,  a  definite  sense  of  progression  along  the  curve  being  taken  as 
positive. 

6.  Velocity  l)eing  defined  as  the  quotient  of  distance  by 
time  in  uniform  motion,  and  as  the  Hmit  of  such  a  quotient 
in  any  motion,  the  unit  of  velocity  is  the  unit  of  length 
divided  by  the  unit  of  time.  Thus  we  speak  of  a  velocity 
of  so  many  centimeters  per  second  (cm./sec),  or  feet  per 
minute  (ft./min.),  or  miles  per  hour  (M./h.),  etc. 

Denoting  the  units  of  time,  length,  and  velocity  by  T,  L, 
V,  this  is  expressed  symbolically  by  writing 

V  =  ^  =  LT-^ 

and  saying  that  the  dimensions  of  velocitj^  (V)  are  1  in 
length  (L)  and  —  1  in  time  (T). 

The  reader  is  supposed  to  be  familiar  with  the  C.G.S. 
(centimeter-gram-second)  and  F.P.S.  (foot-pound-second) 
systems  of  measurement.  It  will  suffice  to  mention  that  the 
second  is  the  -g-g-iFo  P^^t  of  the  mean  solar  day  which  is  the 
average,  for  one  year,  of  the  time  between  two  successive 
passages  of  the  sun  across  the  meridian;  and  that  the  foot 
is  i  of  a  yard,  the  American  yard  being  defined  (by  act  of 


6  KINEMATICS  [7. 

Congress,  1866)  as  |f  |f  of  a  meter.     We  have  therefore  the 

exact  relations 

ft.        1200 

1  cm.  =  0.3937  m.,  =  oTT^, 

cm.      39.37 

which  give  approximately: 

Im.  =  3.2808  ft.,  1  ft.  =  30.48  cm.,  1  in.  =  2.54  cm. 

7.  Exercises. 

(1)  Compare  the  following  velocities  by  reducing  all  to  ft. /sec:  (a) 
man  walking  4  M./h.;  (b)  horse  trotting  a  mile  in  2  min.  10  sec;  (c) 
train  running  40  M./h.;  (d)  ship  making  15  knots,  a  knot  being  a  sea- 
mile  (=  6080  ft.)  per  hour;  (e)  sound  in  dry  air  at  0°  C.  331.3  m./sec; 
(/)  sun  moving  in  space  25  km. /sec;  {g)  light  3  X  10^"  cm. /sec 

(2)  Two  men  starting  (in  opposite  sense)  from  the  same  point  walk 
around  a  block  forming  a  rectangle  of  sides  a,  b;  H  their  constant 
velocities  are  Vi,  V2,  when  and  where  will  they  meet? 

(3)  The  mean  distance  of  the  sun  being  923^  million  miles,  find 
the  velocity  of  light  if  it  takes  light  16  min.  42  sec.  to  cross  the  earth's 
orbit:  (a)  in  miles  per  second,  (fe)  in  kilometers  per  hour. 

(4)  Two  trains,  one  250,  the  other  420  ft.  ^ong,  pass  each  other  on 
parallel  tracks  in  opposite  sense,  with  equal  velocities.  A  passenger 
in  the  shorter  train  observes  that  it  takes  the  longer  train  just  6  sec.  to 
pass  him.     What  is  the  velocity? 

(5)  What  is  the  distance  from  J^  to  5  if  a  man  walking  5  M./h.  can 
cover  it  in  10  min.  less  than  one  walking  3  M./h.? 

(6)  What  is  the  answer  to  the  preceding  problem  if  both  men  start 
from  A  at  the  same  time  and,  when  the  one  has  reached  B,  the  other  is 
7K  miles  behind  him? 

(7)  Two  ships  start  from  the  same  port,  the  second  an  hour  later 
than  the  first.  The  velocity  of  the  first  is  16  knots,  that  of  the  second 
14  knots.  How  many  miles  are  they  apart  3  hours  after  the  first  ship 
started,  the  angle  between  their  paths  being  60°? 

8.  The  definition  of  velocity 

ds 

enables  lis  to  find  the  velocity  when  the  co-ordinate  s  is 


10.1  RECTILINEAR  MOTION   OF  A  POINT  7 

given  as  a  function  of  t.  Conversely,  when  v  is  given  as  a 
function  of  Tor  of  s  (or  of  both  s  and  t),  the  integration  of  the 
same  equation  gives  a  relation  between  s  and  t  which  deter- 
mines the  position  of  the  moving  point  at  any  time. 

Thus,  if  V  is  given  as  a  function  of  t,  we  find  by  integrating 
the  relation  ds  =  vdt: 


—  So  =    I    vdt, 


where  So  is  the  position  of  the  moving  point  at  the  time  to, 
the  so-called  initial  position. 

If  v  is  given  as  a  function  of  s,  we  find  by  integrating  the 
relation  dt  =  ds/v: 

J.s'o       V 

9.  Exercises. 

(1)  Find  the  velocity  when:  (o)  s  =  at  +  h,  (h)  s  =  af  -\-ht  -\-  c, 
(c)  s  =  aVT,  {d)s  =  avoakt,  (r)  s  =  aerf,  (/)  s  =  laic*  +  e-'),  ig)  «  = 
ia(2<3  +  3/2  +  1),  (/i)  s  =  a{C-  -  1)=,  {i)  s  =  af-ii  -  1),  (j)  s  =  a{l^  - 
2W  —  1),  (A-)  s  =  o//(l  —'P).  Taking  a  as  a  positive  constant,  discuss 
the  motion  by  determining  when  s  and  v  have  maxima  or  minima.  The 
nature  of  the  motion  will  be  best  understood  by  skettihing  in  each  case 
the  curve  that  represents  s  as  a  function  of  i,  and  then  imagining  this 
curve  projected  on  the  axis  of  s.  Analytically,  the  sign  of  the  velocity 
determines  the  sense  of  the  motion;  i.  e.  when  v  is  >  0,  s  increases; 
when  V  <  0,  s  decreases;  when  v  —  0  and  dv/dt  4=  0,  «  has  an  extreme 
value  and  the  sense  of  the  motion  changes. 

(2)  Find  the  distance  s  in  terms  of  I  when:  (a)  v  =  vo  +  gl.    (h) 

Sot 

V  =  a(t-  -  4),  (c)  V  =  a  scd^t,  {d)  ?;  =  -    ^     _     ,  {c)  v  =  ac«' '  /3 ;  with 

s  "=  So  for  /  =  0. 

(3)  Find  t  as  a  function  of  s  and  s  as  a  function  of  I  when:  (a)  v  = 
V2gs,  with  s  =  so  for  i  =  0;  {h)  v  =  Va^  -^',  with  s  =  0  when  /  =  0;. 
(c)  V  =  T/a2~+  .s2,  with  s  =  0  for  /  =  0. 

10.  In  reciilinenr  motion,  the  time-rate  of  change  of  the 
velocity  is  called  the  acceleration;  dcMioting  it  by  the  l(>tter  j, 


8  KINEMATICS  [10. 

we  have 

■  _  dv  _  dH 
-^  ~  dt~  dt-' 

We  are  led  to  the  idea  of  acceleration  by  a  process  of  reasoning 
strictly  analogous  to  that  followed  in  defining  velocity  (Ai't.  5).  Among 
non-uniform  motions,  the  most  simple  kind  is  that  in  which  the  velocity 
always  increases  (or  always  decreases)  by  equal  amounts  in  equal  times; 
it  is  called  uniformly  accelerated  motion.  In  this  -kind  of  motion,  the 
quotient  obtained  by  dividing  the  increase  (or  decrease)  of  the  velocity 
in  any  time  by  this  time  is  called  the  acceleration  of  the  uniformly  ac- 
celerated motion. 

If  the  motion  is  not  uniformly  accelerated  the  same  quotient  is  called 
the  average,  or  mean,  acceleration  for  that  time.  Thus,  if  the  velocity 
is  V  at  the  time  t  when  the  moving  point  has  the  position  P,  and  reaches 
the  value  v  +  Aji  at  the  subsequent  time  <  +  A/,  when  the  point  is  at 
P',  the  mean  acceleration  in  the  time  A;  (or  distance  PP'  =  As)#is 
AvjAt.     The  limit  of  this  quotient,  as  At  approaches  zero,  i.  e. 

,.      Av        dv 
•^     At=oAt         dt   ' 

is  called  the  acceleration  at  the  time  t  (or  at  the  distance  s). 

It  follows  that  in  uniformly  accelerated  motion  the  acceleration  is 
constant;  and  conversely,  when  the  acceleration  is  constant,  the  motion 
is  uniformly  accelerated. 

11.  A  rectilinear  motion  is  called  accelerated  whether  the 
velocity  be  increasing  or  decreasing.  But  sometimes  the 
term  acceleration  is  used  in  a  more  restricted  sense,  as  opposed 
to  retardation.  The  motion  is  then  called  accelerated  or 
retarded  according  as  the  absolute  value  of  the  velocity  is 
increasing  or  diminishing.     This  gives  the  criterion 

d(v^)  >  ^       .  dv 

^SO,     ^.<■..J,SO. 

Thus  the  motion  is  accelerated  (in  the  narrower  sense)  or 
retarded  according  as  vdv/dt  is  >  0  or  <  0;  if  dv'dt  =  0  while 


13.]  RECTILINEAR   MOTION   OF  A  POINT  9 

dhjdP  4=  0,  the  motion  changes  from  being  accelerated  to 
being  retarded,  or  vice  versa. 

Acceleration  being  defined,  for  rectilinear  motion,  as  the 
quotient  of  velocity  by  time  or  as  the  limit  of  such  a  quotient, 
the  unit  of  acceleration  J  is  the  unit  of  velocity  divided  by 
the  unit  of  time.  With  the  notation  of  Art.  6,  this  is  ex- 
pressed symbolically  by  writing 

J  =  ^  =  ^  ^  Lr- 

hence  the  dimensions  of  acceleration  are  1  in  length  and  —  2 
in  time.  Thus  we  speak  of  an  acceleration  of  so  many 
centimeters  per  second  per  second  (cm. /sec. ^). 

12.  Exercises. 

(1)  A  point  moving  with  constant  acceleration  gains  at  the  rate  of 
30  M./h.  in  every  minute.     Express  its  acceleration  in  ft./sec.^. 

(2)  At  a  place  where  the  acceleration  of  gravity  is  ^  =  9.810  m./sec.^, 
what  is  the  value  of  g  in  ft. /sec.-? 

(3)  A  railroad  train,  10  min.  after  starting,  attains  a  velocity  of 
45  M./li.;  what  is  its  average  acceleration  during  these  10  min.? 

(4)  How  does  the  acceleration  of  gravity  which  is  about  32.2  ft./sec.^ 
compare  with  that  of  the  train  in  Ex.  (3)? 

(5)  Find  the  acceleration  for  the  motions  in  Art.  9,  Ex.  (1);  apply 
the  rule  of  Art.  11  to  determine  where  each  motion  is  accelerated  or 
retarded. 

(6)  Discuss  in  the  same  way  the  acceleration  of  the  motions  in  Art. 
9,  Ex.  (2)  and  (3). 

13.  A  rectilinear  motion  is  fully  characterized  if  its 
acceleration  is  given  as  a  function  of  t,  s,  v,  provided  that  the 
initial  conditions  are  also  given,  viz.  the  position  and  velocity 
of  the  moving  point  at  any  instant. 

For  we  then  have 

d^s 

-^2=  j(t,s,v),  (1) 


10  KINEMATICS  Fl3. 

where  v  =  ds/dt,  while  j(t,  s,  v)  is  a  given  function.  The 
solution  of  this  differential  equation,  which  is  called  the 
equation  of  motion,  with  the  initial  conditions  s  =  So, 
V  =  I'o  for  t  =  to,  gives  s  as  a  function  of  t. 

The  solution  of  such  a  differential  equation  may  be  diffi- 
cult; nor  can  any  general  rule  of  procedure  be  given.  We 
here  confine  ourselves  to  very  simple  cases,  especially  those 
where  the  acceleration  j  is  either  a  constant  or  a  function 
of  s  alone,  these  cases  being  most  important. 

2.     Examples  of  rectilinear  motion. 

14.  Uniformly  Accelerated  Motion.  As  in  this  case  the 
acceleration  j  is  constant  (see  Art.  10),  the  equation  of  motion 

(1) 

d-s       .  dv 

can  readily  be  integrated: 

V  =jt-^  C. 

To  determine  the  constant  of  integration  C,  we  must  know 
the  value  of  the  velocity  at  some  particular  instant.  Thus, 
\i  V  =  vq  when  t  =  0,  we  find  Vq  =  C;  hence,  substituting 

this  value  for  C, 

V  -  Vo  =  jt.  (2) 

This  equation  gives  the  velocity  at  any  time  t.  Substi- 
tuting ds/dt  for  V  and  integrating,  we  find  s  =  Vot-\-  ^jt^  +  C, 
where  the  constant  of  integration,  C,  must  again  be  deter- 
mined from  given  "  initial  conditions."  Thus,  if  we  know 
that  s  =  So  when  ^  —  0,  we  find  So  —  C;  hence 

s  —  So  ^  Vol  -\-  \jt~.  (3) 

This  equation  gives  the  space  or  distance  passed  over  in 
terms  of  the  time. 


17].  RECTILINEAR   MOTION   OF  A   POINT  11 

Eliminating  j  between  (2)  and  (3),  we  obtain  the  relation 

s  —  So  =  Hvo  +  v)t, 

which  shows  that  in  uniformly  accelerated  motion  the  space  can 
be  found  as  if  it  were  described  uniformly  with  the  mean 
velocity  ^(vo  +  v). 

15.  To  obtain  the  velocity  in  terms  of  the  space,  we 
have  only  to  eliminate  t  between  (2)  and  (3) ;  we  find 

Kv'-vo')  =j{s-So).  (4) 

This  relation  can  also  be  derived  by  eliminating  dt  between 
the  differential  equations  v  =  ds/dt,  dvjdt  =  j,  which  gives 
vdv  =  jds,  and  integrating.  The  same  equation  (4)  is  also 
obtained  directly  from  the  fundamental  equation  of  motion 
d"s/dt-  =  j  by  a  process  very  frequently  used  in  mechanics, 
viz.  by  multiplying  both  members  of  the  equation  by 
ds/dt.  This  makes  the  left-hand  member  the  exact  deriv- 
ative of  ^{ds/dty  or  i^y-,  and  the  integration  can  therefore 
be  performed. 

16.  The  three  equations  (2),  (3),  (4)  contain  the  complete 
solution  of  the  problem  of  uniformly  accelerated  motion.  For 
uniformly  retarded  motion,  j  is  a  negative  number. 

If  the  spaces  be  counted  from  the  position  of  the  moving 
point  at  the  time  f  =  0,  we  have  Sq  =  0,  and  the  equations 
become 

V  =  Vo-\-  jt,     s  =  vd  +  ijt^,     i{v^  -  vo")  =js. 

If  in  addition  the  initial  velocity  Vo  be  zero,  the  point 
starting  from  rest  at  the  time  t  —  0,  the  equations  reduce 
to  the  following: 

V  =  jt,     s  =  ijt-,     iv"^  =  js. 

17.  The  most  important  example  of  uniformly  acceler- 
ated motion  is  furnished  by   a  body  falling  in  vacuo  near 


12  KINEMATICS  [18. 

the  earth's  surface.  Assuming  that  the  body  does  not  rotate 
during  its  fall,  its  motion  relative  to  the  earth  is  a  mere 
translation,  i.  e.  the  velocities  of  all  its  points  are  equal  and 
parallel;  and  it  is  sufficient  to  consider  the  motion  of  any 
one  point  of  the  body.  It  is  known  from  observation  and 
experiment  that  under  these  circumstances  the  acceleration 
of  a  falling  body  is  constant  at  any  given  place  and  equal  to 
about  980  cm.,  or  32.2  ft.,  per  second  per  second;  the  value  of 
this  so-called  acceleration  of  gravity  is  usually  denoted  by  g. 

In  the  exercises  on  falling  bodies  (Art.  19)  we  make  through- 
out the  following  simplifying  assumptions:  the  falling  body 
does  not  rotate;  the  resistance  of  the  air  is  neglected,  or 
the  body  falls  in  vacuo;  the  space  fallen  through  is  so  small 
that  g  may  be  regarded  as  constant;  the  earth  is  regarded 
as  fixed. 

18.  The  velocity  v  acquired  by  a  falling  body  after  falling 
from  rest  through  a  height  h  is  found  from  the  last  equation 
of  Art.  16  as 

V  =  V2gh. 

This  is  usually  called  the  velocity  due  to  the  height  (or 
head)  h,  while  h  =  v'^/2g  is  called  the  height  (or  head)  due 
to  the  velocity  v. 

19.  Exercises. 

(1)  A  body  falls  from  rest  at  a  place  where  g  =  32.2.  Find  (a) 
the  velocity  at  the  end  of  the  fourth  second;  (6)  the  space  fallen  through 
in  4  seconds;  (c)  the  space  fallen  through  in  the  fifth  second. 

(2)  A  train,  starting  from  the  station,  acquires  a  velocity  of  30 
M./h.:  (a)  in  8  min.;  (b)  in  2  miles;  what  was  its  acceleration  (regarded 
as  constant)? 

(3)  Galileo,  who  first  discovered  the  laws  of  falling  bodies,  ex- 
pressed them  in  the  following  form:  (a)  The  velocities  acquired  at 
the  end  of  the  successive  seconds  increase  as  the  natural  numbers; 
(6)  the  spaces  described  during  the  successive  seconds  increase  as  the 


19.J  RECTILINEAR  MOTION   OF  A   POINT  13 

odd  numbers;  (c)  the  spaces  described  from  the  beginning  of  the  motion 
to  the  end  of  the  successive  seconds  increase  as  the  squares  of  the  natural 
numbers.     Prove  these  statements. 

(4)  A  stone  dropped  into  the  vertical  shaft  of  a  mine  is  heard  to 
strike  the  bottom  after  I  seconds;  find  the  depth  of  the  shaft,  if  the 
velocity  of   sound   be   given  =  c.     Assume  <  =  4  s.,  c  =  332   meters,V 
g  =  980.  ^ 

(5)  A  railroad  train  in  approaching  a  station  makes  half  a  mile  in 
the  first;  2,000  ft.  in  the  second,  minute  of  its  retarded  motion.  If 
the  motion  is  uniformly  retarded:  (a)  When  will  it  stop?  (6)  What 
is  the  retardation?  (c)  What  was  the  initial  velocity?  (d)  When 
will  the  velocity  be  4  miles  an  hour? 

(6)  A  body  being  projected  vertically  upwards  with  an  initial  velocity 
vo,  (a)  how  long  and  (6)  to  what  height  will  it  rise?  (c)  When  and 
(d)  with  what  velocity  does  it  reach  the  starting-point? 

(7)  A  bullet  is  shot  vertically  upwards  with  an  initial  velocity  of 
1200  ft.  per  second,  (a)  How  high  will  it  ascend?  (b)  What  is  its 
velocity  at  the  height  of  16,000  ft.?  (c)  When  will  it  reach  the  ground 
again?  ((/)  With  what  velocity?  (e)  At  what  time  is  it  16,000  ft. 
above  the  ground?  Explain  the  meaning  of  the  double  sign  in  (e). 
Use  g  =  32. 

(8)  With  what  velocitj^  must  a  ball  be  tlirown  vertically  upwards  to 
reach  a  height  of  100  ft.? 

(9)  A  body  is  dropped  from  a  point  J5  at  a  height  AB  =  h  above 
the  ground;  at  the  same  time  another  body  is  thrown  vertically  up- 
ward from  the  point  A,  with  an  initial  velocity  ;'o.  (a)  When  and 
(b)  where  will  they  collide?  (c)  If  they  are  to  meet  at  the  height  ^h, 
what  must  be  the  initial  velocity? 

(10)  The  barrel  of  a  rifle  is  30  in.  long;  the  muzzle  velocity  is  1300  ft./ 
sec;  if  the  motion  in  the  baTcl  bo  uniformly  acc(>lerated,  what  is  the 
acceleration  and  what  tlie  lime? 

(11)  If  a  stone  dropped  from  a  balloon  while  ascending  at  the  rate 
of  25  ft. /sec.  reaches  the  ground  in  6  seconds,  what  was  the  height  of 
the  balloon  when  the  stone  was  dropped? 

(12)  If  the  speed  of  a  train  increases  uniformly  after  starting  for  8 
minutes  while  the  train  travels  2  miles,  what  is  the  velocity  acquired? 

(13)  Two  j)articles  fall  from  rest  from  the  same  point,  at  a  .short 
interval  of  time  r;  find  how  far  they  will  be  apart  when  the  first  par- 


14  KINEMATICS  [20. 

tide  has  fallen  through  a  height  h.  Take  e.  g.,  h  =  900  ft.,  t  =  ^^ 
second. 

20.  Acceleration  inversely  proportional  to  the  square  of 
the  distance,  i.  e.  j  =  /x/s^  where  ju  is  a  constant  (viz.  the 
acceleration  at  the  distance  s  =  1)  and  s  is  the  distance  of 
the  moving  point  from  a  fixed  point  in  the  line  of  motion. 

The  differential  equation  (1)  becomes  in  this  case 

the  first  integration  is  readily  performed  by  multiplying  both 
members  by  ds/dt  so  as  to  make  the  left-hand  member  the 
exact  derivative  of  ^{dsjdtY  or  ^v"^.     Thus  we  find 

"r/.s 


^v- 


/^-:+<^'        («) 


where  the  constant  of  integration,  C,  must  be  determined 
from  the  so-called  initial  conditions  of  the  problem.  For 
instance,  if  v  =  v^  when  s  =  So,  we  have  ^V(f  =  —  fx/so  +  C; 
hence,  eliminating  C  between  this  relation  and  (6), 


\S  So  ) 


K„=-.V)  =  -m(^^-^J.  (7) 

To  perform  the  second  integration  solve  this  equation  for 
V  and  substitute  dsldi  for  v. 


di  \j  ''  +  so  ~  s  ' 

or  putting  Vo^  +  2/i/so  =  2)u//i', 

'^^^.  (8) 


-^4'j' 


ds 

dt      "Mm'    N/ 

Here  the  variables  s  and  t  can  be  separated,  and  we  find  if 

5  =  So  for  ^  =  0 


22.]  RECTILINEAR   MOTION  OF  A  POINT  15 


^==^j2'i.X  J,-:^'^^- 


(9) 


To  integrate  put  s  —  /x'  =  x"^.  The  result  will  be  different 
according  to  the  signs  of  /jl,  ix\  and  v,  which  must  be  deter- 
mined from  the  nature  of  the  particular  problem. 

It  is  easily  seen  that  the  methods  of  integration  used  in 
this  problem  apply  whenever  j  is  given  as  a  function  of  s  alone. 

21.  Whenever  in  nature  we  observe  a  motion  not  to  remain  uni- 
form, we  try  to  account  for  the  change  in  the  character  of  tlie  motion 
by  imagining  a  special  cause  for  such  change.  In  rectiUnear  motion, 
the  only  change  that  can  occur  in  the  motion  is  a  change  in  the  velocity, 
i.  e.,  an  acceleration  (or  retardation).  It  ia  often  convenient  to  have 
a  special  name  for  this  supposed  cause  producing  acceleration  or  retarda- 
tion; we  call  it  force  (attraction,  repulsion,  pressure,  tension,  friction, 
resistance  of  a  medium,  elasticity,  cohesion,  etc.),  and  assume  it  to 
be  proportional  to  the  acceleration  A  fuller  discussion  of  the  nature 
of  force  and  its  relation  to  mass  will  be  found  in  Arts.  171-188.  The 
present  remark  is  only  intended  to  make  more  intelligible  the  physical 
meaning  and  application  of  the  problems  to  be  discussed  in  the  follow- 
ing articles. 

22.  It  is  an  empirical  fact  that  the  acceleration  of  bodies  falling  in 
vacuo  on  the  earth's  surface  is  constant  only  for  distances  from  the 
surface  that  are  very  small  in  comparison  with  the  radius  of  the  earth. 
For  larger  distances  the  acceleration  is  found  inversely  proportional  to 
the  square  of  the  distance  from  the  earth's  center. 

By  a  bold  generalization  Newton  assumed  this  law  to  hold  generally 
between  any  two  particles  of  matter,  and  this  assumption  has  been 
verified  by  subsequent  observations.  It  can  therefore  be  regarded 
as  a  general  law  of  nature  that  any  particle  of  matter  produces  in  every 
other  such  particle,  each  particle  being  regarded  as  concentrated  at  a 
point,  an  acceleration  inversely  proportional  to  the  square  of  the  distance 
between  these  points.  This  is  known  as  Neivton's  law  nf  universal  gravi- 
tation, the  acceleration  being  regarded  as  caused  by  a  force  of  attraction 
inherent  in  each  particle  of  matter. 

It  is  shown  in  the  theory  of  attraction  (Art.  253)  that  the  attraction 
of  a  spherical  mass,  such  as  the  earth,  on  any  particle  outside  the  sphere 


16 


KINEMATICS 


[23. 


is  the  same  as  if  the  mass  of  the  sphere  were  concentrated  at  its  center. 
The  acceleration  produced  by  the  earth  on  any  particle  outside  it  is 
therefore  inversely  proportional  to  the  square  of  the  distance  of  the 
particle  from  the  center  of  the  earth. 

23.  Let  us  now  apply  the  general  equations  of  Art.  20  to 
the  particular  case  of  a  body  falling  from  a  great  height 
towards  the  center  of  the  earth,  the 
resistance  of  the  air  being  neglected. 
Let  0  be  the  center  of  the  earth 
(Fig.  3),  Pi  a  point  on  its  surface,  Po 
the  initial  position  of  the  moving 
point  at  the  time  t  =  0,  P  its  posi- 
tion at  the  time  t;  let  OPi  =  R,  OPo 
=  So,  OP  —  s;  and  let  g  be  the  accel- 
eration at  P\,j  the  acceleration  at  P, 
both  in  absolute  value.  Then,  ac- 
cording to  Newton's  law,  j  -.g  =  R-  :s^. 
This  relation  serves  to  determine  the 
value  of  the  constant  fx  in  (5) ;  for  since 
the  acceleration  is  to  have  the  value  g 
when  s  =  R  we  have 


JL 
R^ 


9> 


the   minus   sign   being   taken   because   the   acceleration   is 
directed  toward  the  origin  0.     We  have  therefore 


i"  =  - 

gR\ 

so 

that 

(5)  becomes  in 

our  case 
dt^ 

gR' 

(10) 


24.]  RECTILINEAR   MOTION   OF   A   POINT  17 

the  minus  sign  indicating  that  the  acceleration  tends  to 
diminish  the  distances  counted  from  0  as  origin. 

The  integration  can  now  be  performed  as  in  Art.  20. 
Multiplying  by  dsjdt  and  integrating,  we  find  ^v-  =  gR^js 
+  C.  If  the  initial  velocity  be  zero,  we  have  y  =  Ofors  =  s^; 
hence  C  =  —  gR^/so,  and 

„=_ffv^ JI3=_K J§  N^Z^.    (11) 

\  S        .So  \  So  \         s 

Here  again  the  minus  sign  before  the  radical  is  selected 
since  the  velocity  v  is  directed  in  the  sense  opposite  to  that 
of  the  distance  s. 

Substituting  ds/dt  for  v  and  separating  the  variables  t  and 
s  we  have 

dt  =  -i^^[o"a.| ds; 

R\2g\so  —  s 

hence,  integrating  as  indicated  at  the  end  of  Art.  20: 

the  constant  of  integration  being  zero  since  s  =  So  for  f  =  0. 
The  last  term  can  be  slightly  simplified  by  observing  that 

sin~i  Vl  —  u~  =  cos~Ht, 
whence  finally: 

24.  Exercises. 

(1)  Find  the  velocity  with  which  the  body  arrives  at  the  surface  of 
the  earth  if  it  be  dropped  from  a  height  equal  to  the  earth's  radius,  and 
determine  the  time  of  falling  through   this  heisrht.     Take  R  =  4000 
miles,  g  =  32. 
3 


18  KINEMATICS  [25. 

(2)  Show  that  formula  (11)  reduces  to  v  =  V2gh  (Art.  18)  with 
s  =  R'\i  So  —  s  =  his  small  in  comparison  with  R. 

(3)  Show  that  when  so  is  large  in  comparison  with  R  while  s  differs 
but  sUghtly  from  R,  the  formula;  (11)  and  (12)  reduce  approximately 
to 

, —  R  ,  irsoi 

^^  =  -  V  2o     _,     i  =  ----—  . 
1  s  2V2g  R 

Hence  find  the  final  velocity  and  time  of  fall  of  a  body  falling  to  the 
earth's  surface  (a)  from  an  infinite  distance;  (6)  from  the  moon  (so  = 
mR). 

(4)  Derive  the  expressions  for  v  and  I  corresponding  to  (11)  and 
(12)  when  the  initial  velocity  is  ro  (toward  the  center). 

(5)  A  particle  is  projected  vertically  upwards  from  the  earth's  sur- 
face with  an  initial  velocity  v^.     How  far  and  how  long  will  it  rise? 

(6)  If,  in  (5),  the  initial  velocity  be  ?'o  =  VgR^  how  high  and  how 
long  will  the  particle  rise?  How  long  will  it  take  the  particle  to  rise 
and  fall  back  to  the  earth's  surface? 

25.  Acceleration  directly  proportional  to  the  distance,  i.  e 

j  =  KS,  where  k  is  a  constant. 
The  equation  of  motion 

can  be  integrated  by  the  method  used  in  Art.  20.  The  result 
of  the  second  integration  will  again  be  different  according  to 
the  sign  of  k.  We  shall  study  here  only  a  special  case,  re- 
serving the  general  discussion  of  this  law  of  acceleration  until 
later. 

26.  It  is  shown  in  the  theory  of  attraction  (Art.  251)  that 
the  attraction  of  a  spherical  mass  such  as  the  earth  on  any 
point  within  the  mass  produces  an  acceleration  directed  to 
the  center  of  the  sphere  and  proportional  to  the  distance 
from  this  center.  Thus,  if  we  imagine  a  particle  moving 
along  a  diameter  of  the  earth,  say  in  a  straight  narrow  tube 


26.] 


RECTILINEAR  MOTION   OF   A   POINT 


19 


passing  through  the  center,  we  should  have  a  ease  of  the 
motion  represented  by  equation  (13). 

To  determine  the  value  of  k  for  our  problem  we  notice 
that  at  the  earth's  surface,  that  is,  at  the  distance  OPi  =  R 
from  the  center  0  (Fig.  4),  the 
acceleration  must  be  g.  If,  there- 
fore, j  denote  the  numerical 
value  of  the  acceleration  at  any 
distance  OP  =  s(<  R),  we  have 
j  :  g  =  s  :  R,  or  j  ^  gs/R.  But 
the  acceleration  tends  to  dimin- 
ish the  distance  s,  hence  d^s/dt^ 
=  —  {glR)s.  Denoting  the  posi- 
tive constant  g/R  by  /x^,  the 
equation  of  motion  is 


d  s  I  n 

=  —  /x-s,  where  /*  =  ^f 


dt 


R 


(14) 


Integrating  as  in  Arts.  20  and  23,  we  find 

If  the  particle  starts  from  rest  at  the  surface,  we  have  v  =  0 
when  s  =  R;  hence  0  =  —  ifi^R^  +  C;  and  subtracting 
this  from  the  preceding  equation,  we  find 

V  =  -  IX  V/e^  -  s2,  (15) 

where  the  minus  sign  of  the  square  root  is  selected  because  s 
and  V  have  opposite  sense. 

Writing  dsjdt  for  v  and  separating  the  variables,  we  have 


dt  =  -- 


ds 


M  ^jR^  -  s2' 


whence 


20  KINEMATICS  [27 

jx  K 

As,  s  =  R  when  ^  =  0,  we  have  0  =  ^  cos~i  1  +  C ,  or 
C  =  0.     Solving  for  s,  we  find 

s  =  R  cosfxt.  (16) 

Differentiating,  we  obtain  v  in  terms  of  t: 

V  =  —  fxR  smut.  (17) 

27.  The  motion  represented  by  equations  (16)  and  (17) 
belongs  to  the  important  class  of  simple  harmonic  motions 
(see  Arts.  71  sq.).  The  particle  reaches  the  center  when 
s  =  0,  1.  e.  when  nt  =  -kI'I,  or  at  the  time  t  =  7r/2^t.  At  this 
time  the  velocity  has  its  maximum  value.  After  passing 
through  the  center  the  point  moves  on  to  the  other  end,  P2, 
of  the  diameter,  reaches  this  point  when  s  =  —  R,  i.  e.  when 
p.t  =  TV,  or  at  the  time  t  =  tt/ijl.  As  the  velocity  then  vanishes, 
the  moving  point  begins  the  same  motion  in  the  opposite 
sense. 

The  time  of  performing  one  complete  oscillation  (back  and 
forth)  is  called  the  period  of  the  simple  harmonic  motion; 
it  is  evidently 

T  =  4-—  =  ~. 
2/x       /x  ' 

28.  Exercises. 

(1)  Equation  (14)  is  a  differential  equation  whose  general  integral 
is  known  to  be  of  the  form 

s  =  Ci  sirifit  +  C2  cosfit; 

determine  the  constants  Ci,  C2  and  deduce  equations  (16)  and  (17). 

(2)  Find  the  velocity  at  the  center  and  the  period,  taking  (7  =  32 
and.i2  =  4000  miles. 

(3)  A  point  whose  acceleration  is  proportional  to  its  distance  from 
a  fixed  point  O  starts  at  the  distance  so  from  O  with  a  velocity  Va  directed 
away  from  O;  how  far  will  it  go  before  returning? 


CHAPTER  II. 
TRANSLATION  AND  ROTATION. 

29.  In  kinematics,  the  term  rigid  body  is  used  to  denote  a 
figure  of  invariable  size  and  shape,  or  an  aggregate  of  points 
whose  distances  from  each  other  remain  unchanged. 

The  position  of  a  rigid  body  is  given  if  the  positions  of  any 
three  of  its  points,  not  in  a  straight  Hne,  are  given;  when 
three  such  points  are  fixed  the  body  is  fixed. 

The  kinematics  of  rigid  bodies  will  be  discussed  more  fully 
later  on  (Arts.  114-150);  it  will  here  suffice  to  mention  two 
particular  types  of  motion  of  a  rigid  body:  translation,  and 
rotation  about  a  fixed  axis. 

30.  The  motion  of  a  rigid  body  is  called  a  translation  when 
all  points  of  the  body  describe  equal  and  parallel  curves. 
This  will  be  the  case  if  any  three  points  of  the  body,  not  in  a 
straight  line,  describe  equal  and  parallel  curves.  Owing  to 
the  rigidity  of  the  body,  i.  e.  the  invaria])ility  of  the  mutual 
distances  of  its  points,  the  velocities  and  accelerations  of  all 
points  at  any  given  instant  must  then  be  equal;  thus,  in 
translation,  the  motion  of  the  whole  body  is  given  by  the  motion 
of  any  one  of  its  points. 

31.  When  a  rigid  body  has  two  of  its  points  fixed,  the  only 
motion  it  can  have  is  a  rotation  about  tlu^  line  joining  the 
fixed  points  as  axis.  Thus,  in  rotation  about  a  fixed  axis,  all 
points  of  the  body  excepting  those  on  the  axis  describe  arcs  of 
circles  whose  centers  lie  on  the  axis  and  whose  planes  are 
perpendicular  to  the  axis;  all  points  on  the  axis  are  at  rest. 

The  position  of  a  rigid  ])()dy  wilh  a  fix(^d  axis  I  is  given  by 

21 


22 


KINEMATICS 


(32. 


the  position  of  any  one  of  its  points  P,  not  on  the  axis.  This 
position  is  most  conveniently  assigned  by  tlie  dihedral  angle 
6,  made  by  the  plane  (/,  P)  of  the  body  with  a  fixed  plane 
through  I.  If  a  definite  sense  of  rotation  about  the  axis  is 
assumed  as  positive,  say  the  counter-clockwise  sense  as  seen 
from  a  marked  end  of  the  axis  (Fig.  5), 
the  angle  6,  expressed  in  radians,  is  a  real 
number  and  serves  as  co-ordinate  to  de- 
termine the  position  of  the  body. 

32.  As  the  body  turns  about  the  axis 
I  in  any  way,  the  angle  6  varies  with  the 
time;  the  co-ordinate  d  can  be  regarded 
as  a  function  of  the  time,  just  as  in  the 
case  of  the  rectihnear  motion  of  a  point 
(and  hence  (Art.  30)  also  in  the  rectili- 
near translation  of  a  rigid  body)  the  co- 
ordinate s  is  a  function  of  the  time. 

The  rotation  is  called  uniform  if  equal  angles  are  always 
described  in  equal  times.  In  this  case  the  quotient  Ojt  of 
the  angle  6  described  in  any  time  t,  divided  by  this  time,  is 
called  the  angular  velocity,  co,  of  the  uniform  rotation: 


Fis.  5. 


t  ' 

If,  in  particular,  the  time  of  a  complete  revolution  be 
denoted  by  T,  we  have  for  uniform  rotation: 

27r 

In  the  applications,  angular  velocity  is  often  measured  by 
the  number  of  complete  revolutions  per  unit  of  time.  Thus, 
if  n  be  the  number  of  revolutions  per  second,  A^  that  per 
minute,  we  have* 


34.1  TRANSLATION  AND   ROTATION  23 

CO  =  ZTrn  =  — ^  . 
oU 

33.  When  the  rotation  is  not  uniform,  the  quotient  ob- 
tained by  dividing  the  angle  of  rotation  by  the  time  in  which 
it  is  described,  gives  the  viean,  or  average,  angular  velocity  for 
that  time. 

The  rate  of  change  of  the  angle  of  rotation  with  the  time 

at  any  particular  moment  is  called  the  angular  velocity  at 

that  moment: 

(Id 

The  rate  at  which  the  angular  velocity  changes  with  the 
time  is  called  the  angular  acceleration;  denoting  it  by  a, 
we  have 

"^  ~  dt  ~  dt" 

34.  The  most  important  special  case  of  variable  angular 
velocity  is  that  of  uniformly  accelerated  (or  retarded)  rota- 
tion when  the  angular  acceleration  is  constant.  The  formulae 
for  this  case  have  precisely  the  same  form  as  those  given  in 
Arts.  14-lG  for  uniformly  accelerated  rectilinear  motion. 
Denoting  the  constant  linear  acceleration  by  j,  we  have, 
when  the  initial  velocity  is  0, 


FOR  translation: 

FOR  rotation: 

^  =  j,  a  constant; 

d-'d 

^  =  a,  a.  const; 

V  =  jt, 

o)  =  at. 

s  =  ^3t\ 

6  =  haf". 

W  =  js; 

W   =  ad', 

24-  KINEMATICS  l35. 

and  when  the  initial  velocities  are  Vo  and  coo,  respectively: 
FOR  translation:  for  rotation: 

V  =  Vo  -}-  jt,  CO  =  too  +  Oct, 

S  =  Vot-i-  ijt\  0  =  coo«  +  iar~, 

iy-  —  ho^  =  js;  ico^  —  iwo^  =  ad. 

35.  Let  a  point  P,  whose  perpendicular  distance  fronn  the 
axis  of  rotation  is  OP  =  r,  rotate  about  the  axis  with  the 
angular  velocity  co  =  dd/dt.  In  the  element  of  time,  dt,  it 
will  describe  an  element  of  arc  ds  =  rdd  =  roodt.  Its  velocity 
V  =  ds/dt  (frequently  called  its  linear  velocity  to  distinguish 
it  from  the  angular  velocity)  is  therefore  related  to  the 
angular  velocity  of  rotation  by  the  equation 

V  =  cor. 

The  close  analogy  between  rectilinear  translation  and 
rotation  about  a  fixed  axis,  which  is  not  confined  to  uniform 
or  uniformly  accelerated  motion  and  arises  from  the  fact 
that  in  each  of  the  two  cases  the  position  of  the  body  is 
determined  by  a  single  co-ordinate,  can  be  illustrated  by 
laying  off  on  the  axis  of  rotation  a  length  measuring  the 
angle  of  rotation.  The  rectilinear  motion  of  the  extremity 
of  this  vector  along  the  axis  gives  an  exact  representation 
of  the  rotation. 

36.  Exercises. 

(1)  If  a  fly-wheel  of  10  ft.  diameter  makes  30  revolutions  per  minute, 
what  is  its  angular  velocity,  and  what  is  the  linear  velocity  of  a  point 
on  its  rim? 

(2)  Find  the  constant  acceleration  (such  as  the  retardation  caused 
by  a  Prony  brake)  that  would  bring  the  fly-wheel  in  Ex.  (1)  to  rest  in 
Js  minute.  How  many  revolutions  does  the  fly-wheel  make  during  its 
retarded  motion  before  it  comes  to  rest? 

(3)  A  wheel  is  running  at  a  uniform  speed  of  32  turns  a  second  when 
a  resistance  begins  to  retard  its  motion  uniformly  at  a  rate  of  8  radians 


36.]  TRANSLATION   AND   ROTATION  25 

per  second,     (a)  How  many  turns  will  it  make  before  stopping?     (6)  In 
what  time  is  it  brought  to  rest? 

(4)  A  wheel  of  6  ft.  diameter  is  making  50  rev./min.  when  thrown 
out  of  gear.  If  it  comes  to  rest  in  4  minutes,  find  (a)  the  angular 
retardation;  (o)  the  linear  velocity  of  a  point  on  the  rim  at  the  be- 
ginning of  the  retarded  motion;  (c)  the  same  after  two  minutes. 


CHAPTER  III. 

CURVILINEAR  MOTION  OF  A  POINT. 

1.  Relative  velocity;  composition  and  resolution 
of  velocities. 

37.  It  is  often  convenient  to  think  of  the  velocity  of  a 
point  not  as  a  mere  number,  but  as  a  vector,  i.  e.  a  segment 
PQ  of  a  straight  line  (Fig.  6),  drawn  from  the  point  P  in  the 

direction  of  motion  and  repre- 

'  p  ~^^  senting  by  its  length  the  mag- 

jp-    Q  nitude  of  the  velocity,  by  its 

direction  the  direction  of  mo- 
tion of  P,  and  by  an  arrowhead  the  sense  of  the  motion. 

38.  Consider  a  point  P  (Fig.  7)  moving  along  a  straight 
line  I  with  constant  velocity  Vr,  while  the  line  I  moves  in  a 
fixed  plane  with  a  con- 
stant velocity  Vb  in  a  di- 
rection making  an  angle 
a  with  the  line  I.  Then 
the  vector  PQ  =  Vr  is 
called  the  relative  veloc- 
ity of  P  with  respect  to 
I;  the  vector  PS  =  Vb 
may  be  called  the  body 
velocitij,  or  the  velocity  of  the  body  of  reference  (here  the 
hno  I). 

With  respect  to  the  fixed  plane,  the  point  P  has  not  only 
the  velocity  Vr,  but  it  participates  in  the  motion  of  I.  Its 
absolute  velocihj  v,  i.  e.  its  velocity  with  respect  to  the  fixed 
plane,  is  therefore  represented  in  magnitude,  direction,  and 

26 


39.1  CURVILINEAR  MOTION  OF  A  POINT  27 

sense  by  the  vector  PR,  i.  e.  by  the  diagonal  of  the  parallelo- 
gram constructed  on  the  vectors  Vr  and  I'b.  This  vector 
PR  =  y  is  called  the  resultant,  or  geometric  sum,  of  the  vectors 
PQ  =  Vr  and  PS  =  v^. 

It  is  easy  to  see  that  this  result  will  hold  even  when  the 
motions  are  not  uniform,  provided  we  mean  by  Vr  the  instan- 
taneous relative  velocity  of  P  and  by  Vb  the  simultaneous 
velocity  of  that  point  of  the  body  of  reference  with  which  P 
happens  to  coincide  at  the  instant. 

We  have  thus  the  general  proposition  that  the  absolute 
velocity  v  of  a  point  P  is  the  resultant,  or  geometric  sum,  of  its 
relative  velocity  Vr  and  the  body  velocity  Vb- 

39.  The  term  "  geometric  sum,"  of  the  vectors  Vr  =  PQ  and 
Vb  =  PS  may  be  justified  by  observing  that  (Fig.  7)  QR  = 
PS;  hence  the  resultant  PR  =  i;  is  obtained  simply  by  adding 
the  vectors  Vr  and  Vb  geometrically,  i.  e.  l^y  drawing  first  the 
vector  PQ  =  Vr  and  then  from  its  extremity  Q  the  vector 
QR  =  Vb. 

Conversely,  the  relative  velocity  PQ  =  Vr  is  found  by  geo- 
metrically subtracting  the  body  velocity  Vb  from  the  absolute 
velocity  v;  i.  e.  by  drawing  the  vector  PR  =  ?;,.  and  from  its 
extremity  R  the  vector  RQ  equal  and  opposite  to  the  vector 
PS  =  Vh-  This  result  can  be  interpreted  as  follows:  In  the 
example  of  Art.  38  of  a  point  moving  with  velocity  Vr  along 
the  line  I  while  I  moves  with  velocity  Vb  in  a  fixed  plane,  let  us 
superimpose  the  velocity  —  Vb,  i.  e.  a  velocity  equal  and 
opposite  to  the  body  velocity,  on  the  whole  system,  formed 
by  the  line  and  the  point;  the  line  is  thereby  brought  to  rest 
while  the  point  will  have  the  velocities  v  and  —  Vb  whose 
resultant  is  the  relative  velocity  Vr.  Hence  the  relative 
velocity  is  found  as  the  resultant  of  the  absolute  velocity  and 
the  body  velocity  reversed. 


28 


KINEMATICS 


[40. 


40.  It  is  this  idea  of  relative  motion  that  leads  to  the  so- 
called  parallelogram  of  velocities,  i.  e.  to  the  proposition  that 

a  point  whose  velocity  is  v  =  PR 
(Fig.  8)  can  be  regarded  as  pos- 
sessing simultaneously  any  two 
velocities,  such  as  vt  =  PQ,  v^. 
=  PS  =  QR,  whose  geometric 
sum  is  i;  =  PR.  For  we  can 
always  regard  V\  as  the  relative 
velocity  of  the  point  along  the 

line  PQ  and  V2  as  the  body  velocity,  i.  e.  as  the  velocity  of 

the  line  PQ. 

41.  Finally,  if  in  the  example  of  Art.  38  we  suppose  the 
plane  tt  in  which  the  line  I  moves  to  have  itself  a  velocity  v„, 


Fig.  8. 


Fig.  9. 


it  is  clear  that  the  absolute  velocity  v  of  the  point  will  be 
the  resultant,  or  geometric  sum,  of  the  three  velocities  v,-;  Vb, 
v„;  i.  e.  it  will  be  represented  by  ihe  diagonal  of  the  paral- 
lelepiped that  has  the  vectors  Vr,  Vh,  v„  as  adjacent  edges.  It 
then  follows  that  the  velocity  v  =  PR  of  a  point  (Fig.  9) 
can  be  regarded  as  equivalent  to  any  three  simultaneous 
velocities  Vi  =  PQi,   Vo  =  PQ2,  Vs  =  PQ3,   whose  geometric 


42.]  CURVILINEAR   MOTION   OF  A   POINT  29 

sum  is  V  =  PR.  This  proposition  is  known  as  the  paral- 
lelepiped of  velocities;  Vi,  Vo,  vz  are  called  the  components  of  v. 
The  corresponding  propositions  for  forces  in  statics  will 
be  familiar  to  the  student  from  elementary  physics.  But  it 
will  be  seen  later  that  these  propositions  in  statics  are  really 
based  on  the  more  elementary  propositions  for  velocities. 

42.  Exercises. 

(1)  The  components  of  the  velocity  of  a  point  are  5  and  3  ft. /sec. 
and  enclose  an  angle  of  135°;  find  the  resultant  in  magnitude  and 
direction.     Check  the  result  by  graphical  construction. 

(2)  Find  the  components  of  a  velocity  of  10  ft. /sec,  along  two 
hnes  inclined  to  it  at  30°  and  90°. 

(3)  A  man  jumps  from  a  car  at  an  angle  of  60°,  with  a  velocity  of 
9  ft. /sec.  (relatively  to  the  car).  If  the  car  is  running  10  M./h.,  with 
what  velocity  and  in  what  direction  does  the  man  strike  the  ground? 

(4)  Two  men,  A  and  B,  walking  at  the  rate  of  3  and  4  M./li.,  respec- 
tively, cross  each  other  at  a  rectangular  street  corner.  Find  the  relative 
velocity  of  A  with  respect  to  B  in  magnitude  and  direction. 

(5)  How  must  a  man  throw  a  stone  from  a  train  running  15  M./h, 
to  make  it  move  10  ft. /sec.  at  right  angles  to  the  track? 

(6)  The  velocity  of  light  being  300,000  km. /sec,  the  velocity  of 
the  earth  in  its  orbit  30  km. /sec,  determine  approxunately  the  con- 
stant of  the  aberration  of  the  fixed  stars. 

(7)  A  man  on  a  wheel,  riding  along  the  railroad  track  at  the  rate 
of  9  M./h.,  observes  that  a  train  meeting  him  takes  3  sec  to  pass  him, 
while  a  train  of  equal  length  takes  5  sec.  to  overtake  him.  If  the  trains 
have  the  same  speed,  what  is  it?     What  is  the  length  of  the  train? 

(8)  A  swimmer  starting  from  a  point  A  on  one  bank  of  a  river 
wishes  to  reach  a  certain  point  B  on  the  opposite  bank.  The  velocity 
Vb  of  the  current  and  the  angle  6(  <  I^tt)  made  by  AB  with  the  current 
being  given,  determine  the  least  relative  velocity  v,-  of  the  swimmer  in 
magnitude  and  direction. 

(9)  A  straight  line  in  a  x'lane  turns  with  constant  angular  velocity 
w  about  one  of  its  points  O,  while  a  point  P,  starting  from  0,  moves 
along  the  line  with  constant  velocity  Vo.  Determine  the  absolute  path 
of  P  and  its  absolute  velocity  v. 


30 


KINEMATICS 


[43. 


Fig.  10. 


(10)  Show  how  to  construct  the  tangent  and  normal  to  the  spiral 
of  Archimedes,  r  =  aB,  where  d  =  wt. 

2.  Velocity  in  curvilinear  motion. 

43.  If  on  the  curve  described  by  the  moving  point  we 
select  an  origin  Po,  and  take  a  definite  sense  of  progression 

T  along  the  curve  as  positive, 
the  position  P  of  the  point 
at  any  time  t  is  given  by  the 
arc  PqP  =  s,  which  might 
be  regarded  as  the  co-ordi- 
nate of  P  (Fig.  10). 
As  s  is  a  function  of  the 
time  t,  its  time-derivative 

ds 

'=dt 

gives  the  magnitude  of  the  velocity  of  the  point  at  P,  or  at 
the  instant  t,  in  its  curvilinear  motion  (comp.  Art.  5). 

To  incorporate  in  the  definition  of  velocity  the  idea  of 
the  varying  direction  of  the  motion,  which  at  any  instant  t  is 
that  of  the  tangent  to  the  path,  we  lay  off  from  P,  on  this 
tangent,  a  segment  PT  oi  length  v  =  dsjdt,  in  the  sense  of 
the  motion,  and  define  the  vector  PT  as  the  velocity  of  the 
point  in  its  curvilinear  motion  (comp.  Art.  37). 

44.  When  the  motion  of  the  point  P  is  referred  to  fixed 
rectangular  axes  Ox,  Oij,  Oz,  the  co-ordinates  x,  y,  z  oi  P 
(Fig.  11)  are  functions  of  the  time: 

X  =  x{t),  y  =  y(t),  z  =  z{t). 

Now  the  a:-co-ordinate  of  P  is  at  the  same  time  the  co- 
ordinate of  the  projection  Px  of  P  on  the  axis  Ox  on  this  axis. 
As  the  point  P  moves  in  space,  its  projection  Px  moves 


45. 


CURVILINEAR   MOTION   OF  A   POINT 


31 


along  the  axis  Ox,  and  the  velocity  of  Px  in  its  rectilinear 

motion  is 

dx 


Vx  = 


dt' 


Similarly  the  velocities  of  the  projections  Py,  Pz  of  P  on  Oy, 

Oz  are 

_  dy        _dz 

The  rectilinear  motions  of  P^,  Py,  Pz  along  the  axes  Ox, 
Oy,  Oz,  respectively,  fully  determine  the  curvilinear  motion 
of  P{x,  y,  z)  in  space. 

45.  On  the  other  hand,  the  velocity-vector  PT  =  v  can, 
by  Art.  41,  be  resolved  into  its  three  components  along  the 


Fig.  11. 

axes;  if  the  tangent  to  the  path  at  P  makes  the  angles  a, 
iS,  7  with  Ox,  Oy,  Oz,  respectively,  these  components  are 

V  COSa,    V  COSjS,    V  COS7. 

It  is  easy  to  show  that  these  components  of  v  are  equal, 
respectively,  to  the  velocities  dx/dt,  dy/dt,  dz/dt  of  the  projections 


32  KINEMATICS  [46. 

Px,  Py,  Pz  of  P  on  the  axes.     For  we  have,  if  As  is  the  arc 
described  by  P  in  the  time  A^ : 

dx      ,.       Ax       ,.      Ax  As  ds 

—  =  lim   -—  =  hm  -—  -.^  =  cosa-r;  =  v  cosa, 

dt      A<=o  At       At=o  As  At  at 

since  at  any  ordinary  point  of  the  curve  (i.  e.  at  any  point 
at  which  the  curve  possesses  a  definite  tangent)  we  have 

A^^ 
Um  -7—  =  coso:. 
As 

Similarly  for  dy/dt,  dzldt. 

We  shall  therefore  henceforth  denote  by  v-c,  Vy,  v^  not  only 

(as  in  Art.  44)  the  velocities  of  Px,  Py>  Pz,  but  also  the 

components  of  the  velocity  v  along  the  axes  Ox,  Oy,  Oz. 

Thus  we  have: 

dx  ^       dy  dz 

Vx  =  V  cosa  =  -rr  ,  Vy  =  V  cos/3  =  37  >  ^^  =  i'  COS7  =  ^t", 

In  the  language   of    infinitesimals  we  may  say  that  the 
velocity   is   found   by   dividing   the   element   of   arc  ds  = 
Vdx''-  -\-  dy-  +  dz~  by  dt. 

46.  In  polar  co-ordinates  OP  =  r,  xOP  =  9,  yOQ  =  0  (Fig. 
12),  the  rectangular  components  Vr,  vg,  v^  of  the  velocity  v 
along  OP,  at  right  angles  to'  OP  in  the  plane  xOP,  and  at 
right  angles  to  this  plane  are  readily  found  from  the  last 
remark  in  Art.  45,  by  observing  that 

ds''  =  dr^  +  {rdey  +  (r  sin0  ^0)^, 
whence 

dr  de  ■   n(^ 

'^  =  Jt'''  =  'dt'''='''''^dt' 


47 


CURVILINEAR  MOTION  OF  A  POINT 


33 


Fig.  12. 

47.  If  the  path  of  P  is  a  plane  curve  we  have  in  rectangular 
cartesian  co-ordinates 


dx 


^'~  dt'  ^"  ~  dt' 


dy  ds 


V(l) 


2        /d^J\^ 
■^  ^  dt 


and  in  polar  co-  ordinates 


dr 


do 


ds 


"'  =  If  "' "-'df  "  =  di 


dry        idey 


\dt 


As  the  point  P  moves  in  the  plane  curve  its  radius  vec- 
tor OP  sweeps  out  the  polar  area  S  of  the  curve,  i.  e.  the 
area  bounded  by  any  two  radii  vectores  and  the  arc  of  the 
curve  between  their  ends.  If  AS  be  the  increment  of  this 
area  in  the  time  At,  the  limit  of  the  ratio  AS/At,  as  At  ap- 
proaches zero,  is  called  the  sectorial  velocity  dS/dt  of  the 
point  P  (about  the  origin  0) : 

dS       ,.      AS 
-7-  =  lim    ^^  . 
dt  yt=o  At 

It  follows  from  the  well-known  expression  for  the  element 
of  polar  area  that  in  polar  co-ordinates 


34 


KINEMATICS 


i48. 


dS 
dt 


=   i7- 


dB 
dt' 


and  in  rectangular  cartesian  co-ordinates 


dy 


dx\ 


rf8  ^  ^       _^  _ 
dt   ~  '  \^dt       '^dt  /' 
48.  Exercises. 

(1)  If  the  point  P  describos  a  circle  of  radius  a  about  the  origin  0, 
with  angular  velocity  CO,  the  linear  velocity  of  P  is  /,'  =  oco  (Art.  35);  its 
components  along  rectangular  axes  through  the  Drigin  are  ^Fig.  13): 


Fig.  13. 

Vx  =  aw  cosCiTT  -\-  6)  =  —  aoi  sin0  =  —  uy, 

Vy  =  aoi  sin(2  7r  -\-  0)  =  ow  cosO  =  wx. 

Obtain  these  results  by  differentiating   the   equations   of   the   circle 
X  =  a  cos^,  y  —  a  sinO  with  respect  to  the  time. 

(2)  Show  that  the  velocity  of  a  point  describing  a  cycloid  passes 
through  the  highest  point  of  the  generating  circle. 

(3)  The  ellipse  being  defined  as  the  locus  of  a  point  such  that  the 
sum  of  its  distances  from  two  fixed  points  is  constant;  show  that  the 
normal  bisects  the  angle  between  the  focal  radii  n,  r-i. 

In  bilinear  co-ordinates  the  equation  of  the  ellipse  is  simply 


n  +  r-i  =  2a. 


49.] 


CURVILINEAR   MOTION   OF  A   POINT 


35 


Differentiating  with  respect  to  t  and  denoting  time-derivatives  by  dots, 

we  find 

h  +  h  =  0; 

i.  e.  the  rate  of  increase  of  one  focal  radius  is  equal  to  the  rate  of  de- 
crease of  the  other.  Notice,  however,  that  7\  and  h  are  not  the  com- 
ponents of  the  velocity  of  the  describing  point  P  along  the  focal  radii, 
but  the  projections  of  this  velocity  on  these  radii.  For,  the  velocity 
voi  P  can  be  resolved :  (a)  into  fi  along  ri  and  a  component  perpendicular 
to  n;  (b)  into  r2  along  r2  and  a  component  perpendicular  to  rz.  Both 
resolutions  arise  from  the  sam'e  vector  v;  hence  perpendiculars  erected 
at  the  extremities  of  ri  and  h  (laid  off  from-  P  along  ri,  n  in  the  proper 
sense)  must  meet  at  the  extremity  of  v.  As  rz  =  —  ri,  v  bisects  the 
angle  between  r\  (produced)  and  r2. 

(4)  Find  a  construction  for  the  tangent  to  any  conic  given  by 
directrix,  focus,  and  eccentricity. 

(5)  Derive  the  expressions  for  Vr  and  vq  in  Art.  46  by  the  method 
of  limits. 


3.  Acceleration  in  curvilinear  motion. 

49.  As  the  moving  point  describes  its  path  the  velocity 
vector  V  =  PT  (Art.  43)  will  in  general  vary  both  in  mag- 
nitude and  in  direction.  To  compare  the  velocities  v  =  PT 
at  the  time  t  and  v'  =  P'T'  at  the  time  t  -{-  At  (Fig.  14) 


Fig.  14. 

we  must  draw  these  vectors  from  the  same  origin,  say  from 
the  point  P.     Making  PT"  =  P'T'  =  v',  it  appears  that 


36  KINEMATICS  [50. 

the  vector  v'  can  be  obtained  from  the  vector  v  by  adding 
to  it  geometrically  the  vector  TT"  which  represents  the 
geometrical  increment  of  the  velocity  in  the  time  interval 

This  vector  TT",  divided  by  M,  is  the  average  accelera- 
tion in  the  time  M.  As  M  approaches  zero,  the  vector 
TT"  approaches  zero;  but  its  direction  will  in  general  ap- 
proach a  definite  direction  as  a  limit,  and  the  ratio  of  its 
length  to  M  will  approach  a  definite  number  as  limit.  A 
vector  (generally  drawn  from  the  point  P)  having  this 
limiting  direction  as  its  direction  and  a  length 

rp  rplf 

j  =  lim  -—-— 

''        st=o    At 

is  defined  as  the  acceleration  of  the  moving  point  at  P,  or 
at  the  time  t. 

It  follows  from  this  definition  that  the  acceleration  vector 
lies  in  the  osculating  plane  of  the  path  at  P,  this  plane 
being  the  limiting  position  of  the  plane  determined  by 
the  tangent  at  P  and  any  near  point  P'  of  the  curve  as 
P'  approaches  P  along  the  curve. 

50.  Acceleration  being  defined  as  a  vector  can  be  resolved 
into  components  by  the  parallelogram  or  parallelepiped 
rules  (Arts.  40,  41). 

Thus,  in  particular,  the  acceleration  j,  since  it  lies  in  the 
osculating  plane,  can  be  resolved  into  a  tangential  component 
jt  along  the  tangent,  and  a  normal  component  jn  along  the 
principal  normal  at  P,  the  principal  normal  being  the  inter- 
section of  the  normal  plane  with  the  osculating  plane. 
If  \p  (Fig.  14)  is  the  angle  between  the  velocity  and  the 
acceleration  these  components  are 

jt  =  j  cos;/',    jn  =  j  sinr/'. 


51.]  CURVILINEAR   MOTION   OF  A   POINT  37 

51.  If  from  any  fixed  point  0  we  draw  vectors  OQ  equal 
and  parallel  to  the  velocity  vectors  PT  oi  the  moving  jioint 
P,  the  extremities  Q  lie  on  a  curve  called  the  hodograph 
of  the  path  of  P]  and  it  follows  from  Art.  49  that  the  accel- 
eration vector  of  P  is  equal  and  parallel  to  the  velocity  vector 
in  the  motion  of  Q  along  the  hodograph.  Hence  the  tangential 
and  normal  components  of  the  acceleration  of  P  are  equal, 
respectively,  to  the  components  of  the  velocity'  of  Q  along 
the  radius  vector  OQ  and  at  right  angles  to  it.  Observing 
that  the  acceleration  lies  in  the  osculating  plane  we  have 

therefore  by  Art.  47 

.   _  dv       .    _     dd 
^'  ~  dV    ^''~''dt' 

where  d  is  the  angle  made  by  OQ,  i.  e.  by  the  velocity  vec- 
tor at  P,  with  any  fixed  direction  in  the  osculating  plane. 
Now  if  ds  be  the  element  of  arc  of  the  path  of  P  we  have 
(cbmp.*  below.  Art.  54) 

^  -  1  ^  ^ 

ds~  p'   '  •  vr^^**V^ 

where  p  is.the  radius  of  (first)  curvature  of  the  path  at  P;  hence 

.    _     ddds  _  v^ 
•^"  ~  ^  dsdt   ~  p' 

Thus  we  have  for  the  tangential  acceleration  ji  and  the 
normal  acceleration  j„  of  a  moving  point 

.   _  dv      .    _  v"^ 
^'  ~  dt'    •^"  "  p- 

52.  When  the  rectangular  cartesian  co-ordinates  of  the  ^ 
moving  point  are  given  as  functions  of  the  time, 

X  =  x(t),     y  =  y(t),     z  =  z{t), 

their  first  derivatives  with  respect  to  the  time  are  on  the 


38  KINEMATICS  [52 

one  hand  the  velocities  of  the  projections  Px,  Py,  Pz  of  P 
on  the  axes  in  their  rectihnear  motions,  on  the  other  the 
components  Vx,  Vy,  Vz  of  the  velocity  v  =  dsldt  of  P  in  its 
ciirvilineap  motion  (Art.  44).  Thus,  using  dots  to  denote 
time-derivatives,  we  have 

Vx  =  X,  Vy  =  y  V2  =  z. 

It  will  now  be  shown  that  the  second  lime-derivatives  x,  y,  z 
of  a;,  y,  z,  which  are  the  accelerations  of  Px,  Py,  Pz  in  their 
rectilinear  motions,  are  at  the  same  time  the  components  jx,  jy, 
jz  of  the  acceleration  vector  along  the  axes  of  co-ordinates. 

53.  We  have 

.  _  dx  _  dxds  _    dx 
dt        ds  dt         ds ' 

whence,  differentiating  with  respect  to  /, 

^  ~  dt^  "  dt\7h)  "Itds^^ds'^dt'^ds  "^*^  ds2' 

Writing  down  the  corresponding  expressions  for  y,  z  by 
cyclic  permutation  of  a;,  ?/,  z  we  find : 

.  dx    ,     „  d?-x 

.  dij    ,     „  dhi 

y-'i  +  '-'d^  ■ 

.  dz   ,     .  dh 
ds  as^ 

Now  if  a,  /3,  7  are  the  direction  cosines  of  the  velocity 
vector  we  have 

_  dx     o  _  dy        _  dz 
"  ~  ds'  '^  ~  ts'  ^  ~  ds' 

hence  the  first  terms  in  the  expressions  found  for  x,  ij,  z  are 


54.]  CURVILINEAR  MOTION   OF  A  POINT  39 

the  components  along  the  axes  of  a  vector,  parallel  to  the 
velocity  and  of  length  v,  i.  e.  of  the  tangential  acceleration 
i^Art.  51). 

To  see  that  the  second  terms  are  the  components  of  the 
normal  acceleration  j'„  =  v'^jp  (Art.  51)  we  have  only  to 
remember  that  the  direction  cosines  X,  ix,  v  of  the  principal 
normal  of  any  curve  are 

^  (Px  dry  dh 

^^'ds^'^^'ds^'  '  =  'ds^' 

a  proof  of  this  fact  is  supplied  in  Art.  54 

Thus  it  appears  that  x,  y,  z  are  the  components  along  the 
axes  of  the  total  acceleration  j  of  the  moving  point. 

~r  54.  To  determine  the  (first)  curvature  1/p  and  the  direction  cosines 
X,  fjL,  V  of  the  principal  normal  of  any  curve  imagine  the  curve  described 
by  a  moving  point  P  with  constant  velocity  1.  The  hodograph  con- 
structed at  the  origin  of  co-ordinates,  is  then  a  spherical  curve,  called 
the  spherical  indicatrix,  and  the  co-ordinates  of  the  point  Q  of  this 
indicatrix,  corresponding  to  the  point  P  of  the  given  curve  are  a,  /3,  y. 
Hence,  if  ds'  is  the  element  of  arc  QQ'  of  the  indicatrix  corresponding 
to  the  arc  PP'  —  ds  of  the  given  curve,  we  have 

^  _  (/« d  dx_  _  d-x  ds 

ds'       ds'  ds        ds-  ds' ' 

But  as  the  radii  vectores  of  the  indicatrix  are  parallel  to  the  tangents 
of  the  given  curve  we  have  (Art.  51) 

ds'  ^  1. 

ds  p' 
hence 

.  d^x 

'ds^' 
and  similar  expressions  for  p,  p. 

55.  When  the  path  of  P  is  a  pla7ie  curve  we  have  as  com- 
ponents of  the  acceleration  j  along  rectangular  cartesian  axes 
in  the  plane  of  motion: 


40 


KINEMATICS 


[55. 


df" 


When  polar  co-ordinates  r,  6  are  used  we  may  resolve  the 
acceleration  j  into  a  component  jV  along  the  radius  vector 
OP  =  r  and  a  component  je  at  right  angles  to  r  (Fig.  15). 


Fig.  15. 

They  are  found  by  projecting  jx  =  x  and  jv  =  ^  on  these 
directions.  Differentiating  the  relations  x  =  r  cos6,  y  = 
r  siiid  twice  with  respect  to  t  we  find 

X  =  r  cos^  —  rd  sin0, 
y  =  r  s\n9  +  rd  cos0, 
X  =  (r  -  re^~)  COS0  -  (2fd  +  rd)  sm9, 
y  =  0'  -  rd")  sin0  +  {2fd  +  rd)  cosd. 
These  expressions  show  directly  that 


jr  =  r  -  rd^,    je  =  2rd  +  rd  = 


rdt 


56.  Exercises. 

(1)  Show  that  the  velocity  of  a  moving  point  is  increasing,  con- 
stant, or  diminishing  according  to  the  value  of  the  angle  f  between  v 
and  J  (Fig.  14). 

(2)  Show  that  in  plane  motion  the  sectorial  velocity  (Art.  47)  is 
constant  if  je  —  0,  and  vice  versa. 


56.1 


CURVILINEAR   MOTION   OF  A   POINT 


41 


(3)  Show  that  the  normal  component  of  the  acceleration  is  the 
product  of  the  radius  of  curvature  into  the  square  of  the  angular  velocity 
about  the  center  of  curvature. 

(4)  If  the  acceleration  of  a  point  P  be  always  directed  to  a  fixed 
point  0,  show  that  the  radius  vector  OP  describes  equal  areas  in  equal 
times. 

(5)  Show  that  in  uniform  circular  motion  the  acceleration  is  directed 
to  the  center  and  proportional  to  the  radius. 

(6)  For  motion  in  the  circle  x  =  a  cos9,  y  =  a  sin0  find  jx  and  jy,  jr 
and  jg,  jt  and  _/». 

(7)  A  wheel  rolls  on  a  straight  track;  find  the  acceleration  of  any 
point  on  its  rim,  and  in  particular  that  of  its  lowest  and  highest  points. 

(8)  What  is  the  hodograph  (a)  for  any  rectilinear  motion?  (6)  for 
any  uniform  motion?  (c)  for  uniform  circular  motion?  (d)  What  can 
be  said  about  the  acceleration  of  any  uniform  motion? 

(9)  The  spherical,  or  polar,  co-ordinates  of  a  point  are  the  radius 
vector  r  =  OP  (Fig.  16),  the  polar  distance  or  colatitude  d  =  xOP,  and 


Fig.  16. 


the  longitude  <}>  =  yOQ.  The  cylindrical  co-ordinates  of  the  same  point 
are  r'  =  RP  =  r  sin0,  <^  =  jjOQ,  x  =  QP  =  r  cos0.  Find  the  cylindrical 
components  of  the  acceleration  (along  PP,  normal  to  xOP,  and  along 
QP),  and  hence  show  that  the  spherical  componenti^^long  OP,  per- 
pendicular to  OP  in  the  plane  xOP ,  and  normal  to  xOr)  arc  jr  =  f  — 
rd^  —  r(^2  gin20,  jg  =  rd  -\-  2rd  +  r^-  sin»  cos^,  ./  4,=  r4>  sine  +  2f<^  sinfl 
+  2r(?<^  COS0. 


42  IvINEMATICS  [57. 

57.  The  fundamental  problem  of  the  kinematics  of  the 
point  consists  in  determining  the  motion  of  the  point  when 
the  acceleration  is  given.  In  cartesian  co-ordinates  this 
requires  the  solution  of  the  simultaneous  differential  equa- 
tions 

d~x  _  .       d-y  _  .        d-z  _  . 
dt^  ~  ^"     dt^  ~  ^"^     dt^  ~  ^" 


jx,  jy,  jz  being  given  functions  of  t,  x,  y,  z,  dx/dt,  d.y/dt,  dz/dt. 
A  first  integration  would  give  the  components  of  the  velocity ; 
a  second  integration  should  give  the  co-ordinates  x,  y,  z 
as  functions  of  the  time,  and  hence  also  the  path  of  the 
moving  point. 

It  may  often  be  more  convenient  to  use  polar  co-ordinates; 
in  the  case  of  plane  motion,  we  have  then  the  equations  at 
the  end  of  Art.  55,  with  jr  and  je  as  given  functions  of  t,  r,  6 
and  their  first  time-derivatives. 

If  the  tangential  and  normal  components  of  the  accelera- 
tion are  given  we  can  use  the  equations  (Art.  51) : 

dv  _  .       v^  _  • 

A  number  of  simple  illustrations  will  be  found  in  the 
following  articles. 

4.  Examples  of  curvilinear  motion. 

(a)  Constant  acceleration. 

58.  Motion  on  a  straight  line  under  gravity.  Let  a  point 
P  move  along  a  line  inclined  at  the  angle  6  to  the  horizon, 
under  the  acceleration  g  of  gravity.  The  motion  is  rectilinear; 
the  component  of  the  acceleration  along  the  line  is  g  sin9; 
hence  the  motion  is  uniformly  accelerated.     The  equations 


60.1  CURVILINEAR   MOTION   OF   A   POINT  43 

are  the  same  as  those  for  falhng  bodies  (Arts.  14,  15)  except 
that  g  is  replaced  by  g  sin0. 

A  particle  placed  on  a  smooth  inchned  plane  will  have  this 
motion  if  its  initial  velocity  is  zero  or  directed  along  the 
greatest  slope  of  the  plane. 

59.  Exercises. 

(1)  Show  that  the  final  velocity  is  independent  of  the  inclination; 
in  other  words,  in  sliding  down  a  smooth  inclined  plane  a  body  acquires 
the  same  velocity  as  in  falling  vertically  througli  the  "lieight"  of  the 
plane. 

(2)  Show  that  it  takes  a  body  twice  as  long  to  slide  down  a  plane 
of  30°  inclination  as  it  would  take  it  to  fall  through  the  height  of  the 
plane. 

(3)  At  what  angle  6  should  the  rafters  of  a  roof  of  given  span  2b  be 
inclined  to  make  the  water  run  off  in  the  shortest  time? 

(4)  Prove  that  the  times  of  sliding  from  rest  down  the  chords  issuing 
from  the  highest  (or  lowest)  point  of  a  vertical  circle  are  equal. 

(5)  Show  how  to  construct  geometrically  the  line  of  quickest  (or 
slowest)  descent  from  a  given  point:  (a)  to  a  given  straight  line,  (b)  to  a 
given  circle,  situated  in  the  same  vertical  plane. 

(6)  Analytically,  the  line  of  quickest  or  slowest  descent  from  a  given 
point  to  a  curve  in  the  same  vertical  plane  is  found  by  taking  the 
equation  of  the  curve  in  polar  co-ordinates,  r  =  f{d),  with  the  given 
point  as  origin  and  the  axis  horizontal.  The  time  of  sliding  down  the 
radius  vector  r  is  ;  =  i/2r/{(j  s'm9).  Show  that  this  becomes  a  maximum 
or  minimum  when  tanO  =  f{d)/f'{d),  according  as/(0)  +  f"{e)  is  negative 
or  positive. 

(7)  Show  that  the  line  of  quickest  descent  to  a  parabola  from  its 
focus,  the  axis  of  the  parabola  being  horizontal  and  its  plane  vertical, 
is  inclined  at  60°  to  the  horizon. 

60.  Free  motion  under  gravity.  The  motion  of  a  point, 
when  subject  only  to  the  constant  acceleration  of  gravity  is 
necessarily  in  the  vertical  plane  determined  by  the  initial 
velocity  and  the  direction  of  gravity.  Taking  the  hori- 
zontal line  in  this  plane  through  the  initial  position  0  of  the 


44 


KINEMATICS 


[60. 


point  as  axis  of  x,  and   the  vertical   upwards  as   positive 
axis  of  y  (Fig.  17),  the  components  of  acceleration  along 


Fig.  17. 

these  axes  are  evidently  0  and  —  g,  so  that  the  equations  of 
motion  (Arts.  55,  57)  are 

d;  =  0,     ij  =  -  g. 

The  first  integration  gives 

X  =  ci,     y  =  -  gt  ■{-  Ci. 

To  determine  the  constants  Ci,  Ci  we  must  know  the  initial 
velocity  in  magnitude  and  direction.  If  the  point  starts 
at  the  time  0  from  0  with  a  velocity  ^o,  inclined  to  the  horizon 
at  an  angle  e,  the  angle  of  elevation,  we  have  for  t  =  0: 
X  =  Vo  cose,  y  =  Vo  sine.  Substituting  these  values  we  find 
Ci  =  Vo  cose,  C2  =  Vo  sine,  so  that  the  velocity  components 
at  any  time  t  are : 

X  =  Vo  cose,     y  =  Vo  sine  —  gt. 
Integrating  again  we  find 

X  =  Vo  cose-t,     y  =  sine-^  —  igf^, 

the  constants  of  integration  being  0  since  x  =  0  and  y  =  0 
for  t  =  0. 


61].  CURVILINEAR   MOTION   OF  A  POINT  45 

These  equations  show  that  the  horizontal  projection  of 
the  motion  is  uniform,  while  the  vertical  projection  is  uni- 
formly accelerated,  as  is  otherwise  apparent  from  the  nature 
of  the  problem. 

Eliminating  t  between  the  last  two  equations  we  find  the 
equation  of  the  path 

y  =  tane-o;  -  ^r— ^  -x"^, 

2vo  cos^e 

which  represents  a  parabola  passing  through  the  origin.  To 
find  its  vertex  and  latus  rectum,  divide  by  the  coefficient  of 
x^  and  rearrange: 

X" sme  cose -a;  = cos^e-w; 

g  y 

completing  the  square  in  x,  the  equation  can  be  written  in 
the  form 

[  X  —  ^^  sm2e       =  — cos-e  [  y  —  -^  sm-^e  I . 

\         2g  /  g  V       2g  J 

The  co-ordinates  of  the  vertex  are  therefore  a  =  (yoV26f)sin2e, 
/3  =  (yoV2{/)sin2e;  the  latus  rectum  4a  =  {2vo'^lg)coQ^t;  the 
axis  is  vertical,  and  the  directrix  is  a  horizontal  fine  at  the 
distance  a  =  (?'o^/2g)  cos^e  above  the  vertex. 

61.  Exercises. 

(1)  Show  that  the  velocity  at  any  time  is  w  =  Vv^'^  —  2gy. 

(2)  Prove  that  the  velocity  of  the  projectile  is  equal  in  magnitude 
to  the  velocity  that  it  would  acquire  by  falling  from  the  directrix:  (a) 
at  the  starting  point,  {h)  at  any  point  of  the  path  (see  Art.  18). 

(3)  Show  that  a  body  projected  vertically  upwards  with  the  initial 
velocity  vo  would  just  reach  the  common  directrix  of  all  the  parabolas 
described  by  bodies  projected  at  different  elevations  e  with  the  same 
initial  velocity  !'o. 

(4)  The  range  of  a  projectile  is  the  distance  from  the  starting  point 
to  the  point  where  it  strikes  the  grovmd.  Show  that  on  a  horizontal 
plane  the  range  is  i?  =  2a  =  {vi?l(j)  sin2€, 


46  KINEMATICS  [61. 

(5)  The  lime  of  flight  is  the  whole  time  from  the  beginning  of  the 
motion  to  the  instant  when  the  projectile  strikes  the  ground.  It  is 
best  found  by  considering  the  horizontal  motion  of  the  projectile 
alone,  which  is  uniform.  Show  that  on  a  horizontal  plane  the  time 
of  flight  is  T  =  i2vo/g)  sine. 

(6)  Show  that  the  time  of  flight  and  the  range,  on  a  plane  through 
the  starting  point  incUned  at  an  angle  d  to  the  horizon,  are 

™        2!iosin(e  —  e)            IT,        2ro^6in(e  —  0)cose 
Tq  =  —  ■ — - ,    and     Re  =  —  • ;:. • 

(7)  What  elevation  gives  the  greatest  range  on  a  horizontal  plane? 

(8)  Show  that  on  a  plane  rising  at  an  angle  d  to  the  horizon,  to 
obtain  the  greatest  range,  the  direction  of  the  initial  velocity  should 
bisect  the  angle  between  the  plane  and  the  vertical. 

(9)  A  stone  is  dropped  from  a  balloon  which,  at  a  height  of  625  ft., 
is  carried  along  by  a  horizontal  air-current  at  the  rate  of  15  miles  an 
hour,  (o)  Where,  (6)  when,  and  (c)  with  what  velocity  will  it  reach 
the  ground? 

(10)  What  must  be  the  initial  velocity  ro  of  a  projectile  if,  with  an 
elevation  of  30°,  it  is  to  strike  an  object  100  ft.  above  the  horizontal 
plane  of  the  starting  point  at  a  horizontal  distance  from  the  latter  of 
1200  ft? 

(11)  Whsit  must  be  the  elevation  e  to  strike  an  object  100  ft.  above 
the  horizontal  plane  of  the  starting  point  and  5000  ft.  distant,  if  the 
initial  velocity  be  1200  ft.  per  second? 

(12)  Show  that  to  strike  an  object  situated  in  the  horizontal  plane 
of  the  starting  point  at  a  distance  x  from  the  latter,  the  elevation  must 
be  €  or  90°  —  e,  where  t  =  J  sin"'  (gx/vo^). 

(13)  The  initial  velocity  Vo  being  given  in  magnitude  and  direction, 
show  how  to  construct  the  path  graphically. 

(14)  The  solution  of  Ex.  (11)  shows  that  a  point  that  can  be  reached 
with  a  given  initial  velocity  can  in  general  be  reached  by  two  different 
elevations.  Find  the  locus  of  the  points  that  can  be  reached  by  only 
one  elevation,  and  show  that  it  is  the  envelope  of  all  the  parabolas 
that  can  be  described  with  the  same  initial  velocity  (in  one  vertical 
plane). 

(15)  If  it  bo  known  that  the  path  of  a  point  is  a  parabola  and  that 


63.J  CURVILINEAR  MOTION  OF  A  POINT  47 

the  acceleration  is  parallel  to  its  axis,  show  that  the  acceleration  is 
constant. 

(16)  Prove  that  a  projectile  whose  elevation  is  60°  rises  three  times 
as  high  as  when  its  elevation  is  30°,  the  magnitude  of  the  initial  velocity 
being  the  same  in  each  case. 

(17)  Construct  the  hodograph  for  the  motion  of  Art.  60,  taking  the 
focus  as  pole  and  drawing  the  radii  vectores  at  right  angles  to  the 
velocities. 

(IS)  A  stone  slides  down  a  roof  sloping  30°  to  the  horizon,  through 
a  distance  of  12  ft.  If  the  lower  edge  of  the  roof  be  50  ft.  above  the 
ground,  (o)  when,  (6)  where,  (c)  with  what  velocity  does  the  stone 
strike  the  ground? 

(19)  If  a  golf  ball  be  driven  from  the  tee  horizontally  with  initial 
speed  =  300  ft. /sec,  where  and  when  would  it  land  on  ground  16  ft. 
below  the  tee  if  resistance  of  air  and  rotation  of  ball  could  be  neglected? 

(20)  A  man  standing  15  ft.  from  a  pole  150  ft.  high  aims  at  the  top 
of  the  pole.  If  the  bullet  just  misses  the  top  where  will  it  strike  the 
ground  if  vo=  1000  ft. /sec? 

62.  While  the  type  of  motion  discussed  in  Art.  60  is  commonly  spoken 
of  as  projectile  moliori,  it  should  be  kept  in  mind  that  it  takes  no  account 
of  the  resistance  of  the  air;  it  gives  the  motion  of  a  projectile  in  vacuo. 
Owing  to  the  very  high  initial  velocities  of  modern  rifle  bullets,  the 
range  may  be  only  about  one  tenth  of  what  it  would  be  according  to 
the  formula?  given  above. 

The  study  of  the  actual  motion  of  a  projectile  in  a  resisting  medium, 
such  as  air,  forms  the  subject  of  the  science  of  ballistics.  See  for 
instance  C.  Cranz,  Lehrbuch  der  BalUstik,  Vol.  I,  2te  Auflage,  Leipzig, 
Teubner,  1910. 

(b)  The  pendulum. 

63.  The  mathematical  pendulum  is  a  point  constrained  to 
move  in  a  vertical  circle  under  the  acceleration  of  gravity. 

Let  0  be  the  center  (Fig.  18),  A  the  lowest,  and  B  the 
highest  point  of  the  circle.  The  radius  OA  =  I  oi  the  circle 
is  called  the  length  of  the  pendulum.  Any  position  P  of 
the  moving  point  is  determined  by  the  angle   AOP  =  6 


48 


KINEMATICS 


[64. 


counted  from  the  vertical  radius  OA  in  the  positive  (counter- 
clockwise) sense  of  rotation. 

If  Pq  be  the  initial  position  of  the  moving  point  at  the 
time  t  =  0,  and  2^  AOPo  =  do,  then  the  arc  PoP  =  s  de- 
scribed in  the  time  ^  is  s  = 
^(^0  —  d);  hence  v  =  ds/dt  — 
-  Idd/dt, and  dv/dt  =  -  Id^d/dt^, 
the  negative  sign  indicating 
that  6  diminishes  as  s  and  t 
increase. 

Resolving  the  acceleration 
of  gravity,  g,  into  its  normal 
and  tangential  components 
g  COS0,  g  sin^,  and  considering 
that  the  former  is  without 
effect  owing  to  the  condition 
that  the  point  is  constrained 
to  move  in  a  circle,  we  obtain  the  equation  of  motion  in 
the  form  dv/dt  =  g  sin0,  or 

.d'9 


Fig.  18. 


mg 


J^  +  <,  sine  =  0. 


(1) 


64.  The  first  integration  is  readily  performed  by  multiply- 
ing the  equation  by  dd/dt  which  makes  the  left-hand  member 
an  exact  derivative, 

dt 


[l(^)   "''"'^1' 


hence  integrating,  we  obtain 

dd 
dt 


U 


—  g  COS0  =  C, 


or  considering  that  v  =  —  Idd/dt, 


65.]  CURVILINEAR  MOTION  OF  A  POINT  49 

^v^  —  gl  COS0  =  CI. 

To  determine  the  constant  C,  the  initial  velocity  Vq  at  the 
time  t  =  0  must  be  given.  We  then  have  i^o"  —  gl  cos^o  = 
CI;  hence 

|y2    =    ^^^2    _    gl  COS0O  +   gl  COS0 

'vq"^  \  (2) 

—  I  COS^o  +  I  COS0 


.2^ 

The  right-hand  member  can  readily  be  interpreted  geo- 
metrically; ^'o^/2gr  is  the  height  by  falling  through  which  the 
point  would  acquire  the  initial  velocity  Va  (see  Art.  18); 
I  COS0  —  I  cos^o  =  OQ  —  OQq  =  QoQ,  if  Q,  Qo  are  the  pro- 
jections of  P,  Po  on  the  vertical  AB.  If  we  draw  a  hori- 
zontal line  MN  at  the  height  vo'^/2g  above  Po  and  if  this 
line  intersect  the  vertical  AB  at  R,  we  have  for  the  velocity 
V  the  expression: 

h'  =  g-RQ. 
If  the  initial  velocity  be  zero,  the  equation  would  be 
h'  =  g-QoQ. 

At  the  points  M,  N  where  the  horizontal  line  MN  inter- 
sects the  circle  the  velocity  becomes  zero.  The  point  can 
therefore  never  rise  above  these  points. 

Now,  according  to  the  value  of  the  initial  velocity  Vo,  the 
line  AIN  may  intersect  the  circle  in  two  real  points  M,  N, 
or  touch  it  at  B,  or  not  meet  it  at  all.  In  the  first  case  the 
point  P  performs  oscillations,  passing  from  its  initial  position 
Po  through  A  up  to  M,  then  falling  back  to  A  and  rising  to 
A'',  etc.     In  the  third  case  P  makes  complete  revolutions. 

65.  The  second  integration  of  the  equation  of  motion 
cannot  be  effected  in  finite  terms,  without  introducing  elliptic 
functions.  But  for  the  case  of  most  practical  importance, 
5 


50  KINEMATICS  1 66. 

viz.  for  very  small  values  of  6,  it  is  easy  to  obtain  an  ap- 
proximate solution.  In  this  case  6  can  be  substituted  for 
sin0,  and  the  equation  becomes: 


or,  putting  g/l  =  m" 


f  =  -  "'»■  ^  (3) 

This  is  a  well  known  differential  equation  (compare  Art. 
26,  eq.  (14),  and  Art.  28,  Ex.  1),  whose  general  integral  is 

6  =  Ci  cos/if  +  C2  sin/jLt. 

The  constants  Ci,  Co  can  be  determined  from  the  initial 
conditions  for  which  we  shall  now  take  6  =  60  and  v  =  0 
when  ^  =  0;  this  gives  Ci  =  60,  Co  =  0;  hence 

1  0 

6  =  60  cosfj.t,  t  =  -  cos"^  — . 

The  last  equation  gives  with  6  =  —  60  the  time  ti  of  one 
swing  or  beat,  that  is,  half  the  period: 

^.-"  =  .rJl  ■  (4) 

M  \g 

The  time  of  a  small  oscillation  or  swing  is  thus  seen  to  be 
independent  of  the  arc  through  which  the  pendulum  swings; 
in  other  words,  for  all  small  arcs  the  times  of  swing  of  the 
same  pendulum  are  very  nearly  the  same;  such  oscillations 
are  therefore  called  isochronous. 

66.  The  formula  (4)  shows  that  for  a  pendulum  of  given  length  h 
the  time  of  one  swing  /i  varies  for  different  places  owing  to  the  variation 
of  g.  As  h  and  'i  can  be  measured  very  accurately,  the  pendulum  can 
be  used  to  determine  g,  the  acceleration  of  gravity  at  any  place;  (4) 
gives : 


67.]  CURVILINEAR   MOTION   OF  A  POINT  51 

Now  let  lo  be  the  length  of  a  pendulum  which  beats  seconds,  i.  e., 
makes  just  one  swing  per  second;  by  (4)  and  (5)  we  find  for  the  length 
lo  of  such  a  seconds  pendulum: 

'o  =  ^2  =  ry  •  (6) 

The  length  k  of  the  seconds  pendulum  is  therefore  found  by  measuring 
the  length  h  and  the  time  of  swing  ti  of  any  pendulum.  This  length  k 
is  very  nearly  a  meter;  it  varies  sUghtly  with  g;  thus,  for  points  at  the 
sea  level  it  varies  from  ^o  =  99.103  cm.  at  the  equator  to  lo  —  99.610 
at  the  poles. 

If  go  be  the  value  of  g  at  sea  level,  i.  e.,  at  the  distance  R  from  the 
center  of  the  earth,  gi  the  value  of  g  at  an  elevation  h  above  sea  level 
in  the  same  latitude,  it  is  known  that 

fir„  ^   {R  +  hY 

gi  R'       ' 

Hence,  if  go  be  known,  pendulum  experiments  might  serve  to  find  the 
altitude  of  a  place  above  sea  level;  but  the  observations  would  have 
to  be  of  very  great  accuracy. 

67.  Let  n  be  the  number  of  swings  made  by  a  pendulum  of  length 
I  in  any  time  T  so  that  h  =  T/n.     Then,  by  (4), 

T  7 

If  T  and  one  of  the  three  quantities  n,  I,  g  in  this  equation  be  re- 
garded as  constant,  the  small  variations  of  the  two  others  can  be  found 
approximately  by  differentiation.  For  instance,  if  the  daily  number 
of  oscillations  of  a  pendulum  of  constant  length  be  observed  at  two 
different  places,  T  and  I  keep  the  same  values  while  n  and  g  vary  by 
small  amounts,  say  An  and  Ag.     Now  the  differentiation  of  (7)  gives 


or,  dividing  by  (7) : 


T,  TvVldg 

-n^^''=--~2    gl 

dn  _  J  dg 
n   ~       g  ' 


We  have  therefore  approximately,  for  small  variations  An,  Ag: 

^^i,^^.  (8) 

n        '   g 


52  KINEMATICS 


168. 


68.  Exercises. 

(1)  Find  the  number  of  swings  made  in  a  second  and  in  a  day  by 
a  pendulum  1  meter  long,  at  a  place  where  g  =  980.5. 

(2)  Find  the  length  of  the  seconds  pendulimi  at  a  place  where  g  = 
32.17. 

(3)  Find  the  value  of  gr  at  a  place  where  a  pendulum  of  length 
3.249  ft.  is  found  to  make  86522  swings  in  24  hours. 

(4)  A  chandelier  suspended  from  the  ceiling  is  seen  to  make  20 
swings  a  minute;  find  its  distance  from  the  ceiling. 

(5)  A  pendulum  of  length  1  meter  is  carried  from  the  equator 
where  g  =  978.1  to  another  latitude;  if  it  gains  100  swings  a  day 
find  the  value  of  g  there. 

(6)  Investigate  whether  the  approximate  formula  (8)  is  sufficiently 
accurate  for  Ex.  (5). 

(7)  If  the  length  of  a  pendulum  be  increased  by  a  small  amount 

Al,  show  that  the  daily  number  of  swings,  n,  will  be  diminished  by 

An  so  that  approximately 

An  _       J  AZ 

(8)  A  clock  beating  seconds  is  gaining  5  minutes  a  day;  how  much 
should  the  pendulum  bob  be  screwed  up  or  down? 

(9)  A  clock  beating  seconds  at  a  place  where  g  =  32.20  is  carried 
to  a  place  where  g  =  32.15;  how  much  will  it  gain  or  lose  per  day  if 
the  length  of  the  pendulum  be  not  changed? 

(10)  A  pendulum  of  length  100.18  cm.  is  foimd  to  beat  3585  times 
per  hour;  find  the  elevation  of  the  place  if  in  the  same  latitude  g  = 
981.02  at  sea  level. 

69.  When  the  oscillations  of  a  pendulum  are  not  so  small 
that  the  angle  can  be  substituted  for  its  sine,  as  was  done  in 
Art.  65,  an  expression  for  the  time  h  of  one  swing  can  be 
obtained  as  follows. 

We  have  by  (2),  Art.  64. 

iw^  —  ivo^  =  gl{cosd  —  cos^o). 

Let  the  time  be  counted  from  the  instant  when  the  moving 
point  has  its  highest  position  {N  in  Fig.  18),  so  that  vo  =  0. 


69. 


CURVILINEAR  MOTION   OF  A   POINT  53 


Substituting  v  =  —  Idd/dt  and  applying  the  formula  cos6  = 
1  —  2  sin^i^  we  find : 

il(§Y  =  2?(sin2i5o  -  sin^ie), 

whence 

.      Pf  dd 

dt 


■*i^ 


g  /sin^l-^o  -  sin^^^  ' 
Integrating  from  9  =  0  to  6  ^  do  and  multiplying  by  2 
we  find  for  the  time  h  of  one  swing: 
I   r^«  dd 


h  = 


Vli 


g  Jo    Vsin^i^o  —  sin^^-^  * 
As  6  cannot  become  greater  than  Oq  we  may  put  sini^  = 
sini^o  sin0,  thus  introducing  a  new  variable  <^  for  which  the 
limits  are  0  and  7r/2.     Differentiating  the  equation  of  substi- 
tution, we  have 

i  cos^O  dd  =  sini^o  cos0  d(f), 


or,  as  cosi^  =  VI  —  sin^i^o  sm^^, 

2  sini^o  coS(/>  dcj) 


dd  = 


V  1  —  sin'-i^o  sinV 
Substituting  these  values  and  putting  for  the  sake  of  brevity 


sini^o  =  K, 


we  find  for  the  time  ti  of  one  swing: 

d(j) 


ti 


■'M 


g  Jo  Vl  —  K^  sin^^ 
The  integral  in  this  expression  is  called  the  complete  elliptic 
integral  of  the  first  species  and  is  usually  denoted  by  K.  Its 
value  can  be  found  from  tables  of  elliptic  integrals  or  by 
expanding  the  argument  into  an  infinite  series  by  the  binomial 
theorem  (since  k  sin0  is  less  than  1 ) ,  and  then  performing  the 


54  KINEMATICS  [71. 

integration.     We  have 


1-3 

(1  -  K^  sin2</>)-J  =  1  +  i/c^sinV  +  — ^  k'  sinV  + 

2-4 


hence 


^'-4b^(Xf--{m^--} 


If  H  be  the  height  of  the  initial  point  N{d  =  6o)  above  the 
lowest  point  A  of  the  circle,  we  have 

2        .   ^1 .        1  —  cos^o      H 
,^  =  ,n,-^0o= =  -^, 

so  that  the  expression  for  ti  can  be  written  in  the  form 

70.  Exercises. 

(1)  Show  that  /.  =  TT  ;/r/^(l  +  Jj  +  To':ri  +  tgVjt  +  • '  )  if  the 
angle  20o  of  the  swing  is  120°. 

(2)  Show  that  as  second  approximation  to  the  time  of  a  small  swmg 
we  have  h  =  irVllffil  +  tV^o")- 

(3)  Find  the  time  of  oscillation  of  a  pendulum  whose  length  is  1 
meter  at  a  place  where  (j  =  980.8,  to  four  decimal  places,  the  amplitude 
Oa  of  the  swing  being  6°. 

(4)  Denoting  bj^  /o  the  first  approximation,  irVl/g,  to  the  time  h 
of  one  swing,  the  quotient  (U  —  to)/io  is  called  the  correction  for  mnplilude. 
Show  that  its  value  is  0.0005  for  do  =  5°. 

(5)  A  pendulum  hanging  at  rest  is  given  an  initial  velocity  vu  Find 
to  what  height  hi  it  will  rise. 

(6)  Discuss  the  pendulum  problem  in  the  particular  case  when  MA'' 
(Fig.  18)  touches  the  circle  at  B,  that  is  when  the  initial  velocity  is 
due  to  falling  from  the  highest  point  of  the  circfle. 

(c)  Simple  harmonic  motion. 

71.  Simple  harmonic  motion  is  that  kind  of  rectilinear 
motion  in  which  the  acceleration  is  proportional  to  the  dis- 
tance of  the  moving  point  P  from  a  fixed  point  0  in  the 


72.]  CURVILINEAR  MOTION   OF  A  POINT  55 

line  of  motion  and  is  always  directed  toward  this  fixed  point 
(Fig.  19). 

An  example  of  simple  harmonic  motion  was  discussed  in 
Arts.  26,  27.  We  now  resume  its  study  from  a  more  general 
point  of  view,  owing  to  its  great  importance.     It  naturally 

P 

1 1 ^_ — ^_^ 

Pg  0  .?  Pi  *■ 

Fig.  19. 

leads  to  the  study  of  certain  important  motions  known  as 

coinpound  harmonic,  which  may  be  curvilinear. 

By  definition,  the  differential  equation  of  simple  harmonic 

motion  is 

X  =   —  fi-X, 

where  ju  is  a  constant,  /x^  being  evidently  the  absolute  value 

of  the  acceleration  at  the  distance  x  =  1  from  the  origin  0. 

The  equation  has  the  form  of  the  pendulum  equation  (3), 

Art.  65,  except  that  6  is  replaced  by  x.     Its  general  integral 

is  therefore 

X  =  Ci  coS)u/  +  C2  smut. 

Differentiating,  we  find  the  velocity 

V  =  —  CiM  ^in/jLt  +  C2M  cosnt. 

If  a:  =  a;o  and  i^  =  i^o  for  i  =  0  we  find  Ci  =  Xo,  C2  =  Vq/ij.', 
hence 

X  =  Xo  cosfit  -\ —  sinut,     v  ^  —  x^p.  sin/xi  +  vo  cosut. 

72.  The  expression  found  for  x  can  be  given  a  more  con- 
venient form  by  observing  that  if  we  construct  a  right-angled 
triangle  (Fig.  20)  with  xo  and  Voffx  as  sides  and  call  a  its 
hypotenuse,  e  its  angle  adjacent  to  Xo,  we  have 


56  KINEMATICS  1 72. 

Xo  =  a  cose,    —  =  a  sine: 

substituting  these  values  we  find 

X  =  a  cose  cos/if  +  sine  sinjui 
=  a  cos(fjLt  —  e). 

Hence,  in  simple  harmonic  motion  we  have 

X  =  a  cos(ixt  —  e),     V  =  —  ajx  sin()uf  —  e), 
where 


■=J 


xo-  +-T>    ^  "=  tan  1 -. 


The  motion  is  clearly  periodic  since  both  position  and  velocity 
regain  the  same  values  when  the  angle  jui  —  e  is  increased  by 
any  integral  multiple  n  of  2t,  i.  e.  if  the  time  t  is  increased  by 
n  times  27r/ju.     The  time 

J" 

between  any  two  successive  equal  stages  of  the  motion  is 
called  the  period;  the  length  a,  which  is  evidently  the 
greatest  distance  on  either  side  of  the  origin  reached  by  the 
point,  is  called  the  amplitude  of  the  simple  harmonic  motion. 

The  angle  fxt  —  e  \s  called  the  phase-angle,  e  the  epoch- 
angle  of  the  motion. 

The  point  oscillates  between  the  positions  Pi  and  P2  (Fig. 
19)  whose  abscissas  are  =ta.  It  is  at  Pi  (at  elongation)  at  the 
time  ^0  =  e//x  (and  also  at  the  times  to  +  n-  27r//x  =  (e  +  2mr)j  p) ; 
it  reaches  the  position  0  at  the  time  fi  =  (e  +-2-7r)/ju,  so  that 
the  time  of  passing  from  Pi  to  0  is 

The  time  of  passing  from  0  to  the  other  elongation  P2  is 


73.] 


CURVILINEAR  MOTION   OF   A   POINT 


57 


easily  shown  to  be  equal  to  this;  so  that  the  time  of  one 
swing  (from  Pi  to  P2)  is 


The  backward  motion  from  P2  to  Pi  takes  place  in  the  same 
time  so  that  the  period,  that  is  the  time  of  a  double  (forward 
and  backward)  swing,  is,  as  shown  above, 

T  =  ^ 

73.,  An  instructive  illustration  is  obtained  by  observing  that  any 
simple  harmonic  motion  can  he  regarded  as  the  'projection  of  a  uniform 
circular  motion  on  a  diameter  of  the  circle.  In  other  words,  it  is  the 
apparent  motion  of  a  point  describing  a  circle  uniformly,  as  seen  from 
a  point  in  the  plane  of  the  circle  (at  an  infinite  distance).  For,  let  a 
point  Q  (Fig.  21)  describe  a  circle  of  radius  a  with  constant  angular 


Fig.  21. 

velocity  w,  say  in  the  counterclockwise  sense.  If  Qo  is  the  position  of 
the  point  at  the  time  i  =  0,  we  have  QoOQ  =  oil,  so  that  the  projection 
of  Q  on  the  diameter  OQa  has,  for  the  center  O  as  origin,  the  abscissa 


58  KINEMATICS  [74. 

a  coswt.  And  if  P  be  the  projection  of  Q  on  a  diameter  OA  making 
with  OQo  the  angle  e,  the  abscissa  of  P  will  be 

X  =  a  cos{u}i  —  e). 

Hence  the  motion  of  P  is  a  simple  harmonic  motion  for  which  the 
acceleration  at  unit  distance  from  0  is  m^  =  t^^- 

74.  Notice  that  the  linear  velocity  v  =  ace  oi  Q  has  along  OA  the 

component 

Vz=  X  =  —aw  sin(co/  —  e), 

which  is  the  velocity  of  P;  and  the  acceleration  of  Q,  j  =  ow^  along  QO, 
has  along  OA  the  component 

jx  =  x  =  —  aw^  cos(w/  —  e)  =  —  w^x, 

which  is  the  acceleration  of  P. 

The  projection  of  the  uniform  circular  motion  of  Q  on  the  diameter 
OB,  perpendicular  to  OA,  gives  also  a  simple  harmonic  motion,  viz. 

y  =  a  sin(aj<  —  t)  =  a  cos[co/  —  (e  +  lir)\, 

which  merely  differs  by  Itt  in  phase  from  the  motion  along  OA. 

The  period  of  the  simple  harmonic  motion  of  P  along  OA  is  (Art.  72) : 

T  =  litjix), 

i.  e.,  it  is  equal  to  the  time  in  which  Q  makes  one  revolution  on  the 
circle.  The  fact  that  this  period  depends  only  on  the  angular  velocity 
and  not  on  the  radius  a,  i.  e.  on  the  amplitude,  is  expressed  by  saying 
that  simple  harmonic  motions  of  the  same  ^  or  w  are  isochronous. 

If  Q  describes  the  circle  p  times  per  second  so  that  P  makes  p  com- 
plete (forth  and  back)  oscillations  per  second,  we  have  w  =  27rp,  so  that 

T  =  1/p; 

i.  e.  the  number  of  oscillations  per  second,  the  so-called  frequency,  is  the 
reciprocal  of  the  period. 

75.  Exercises. 

(1)  Integrate  the  equation  x  =  —  \iH,  by  multiplying  it  by  x,  and 
determine  the  constants  of  integration  if  x  =  xo,  w  =  Vq  for  t  =  0. 

(2)  Show  that  the  period  T  can  be  expressed  in  the  form  li^V  —  x/x; 
also  find  the  velocity  in  terms  of  x. 


77.]  CURVILINEAR  MOTION  OF  A  POINT  59 

(d)  Compound  harmonic  motion. 

76.  Apart  from  the  initial  conditions,  a  simple  harmonic 
motion  is  fully  determined  by  its  line  I,  its  center  0,  and  its 
period  (or  frequency) ,  which  determines  the  constant  fx.  The 
amplitude  a  and  the  phase  e  depend  on  the  initial  conditions 
(see  Art.  72). 

Let  a  point  P  have  a  simple  harmonic  motion  of  period 
T  =  2t/ijl  along  a  line  I,  about  the  center  0;  and  let  the 
line  I  have  a  motion  of  rectilinear  translation  in  a  fixed  plane 
TT  (comp.  Art.  38).  If  the  motion  of  I  is  likewise  a  simple 
harmonic  motion,  al^out  0  as  center,  in  a  direction  U,  the 
absolute  motion  of  P  in  the  plane  tt  is  called  a  compound 
harmonic  motion.  This  is  in  general  a  curvilinear  motion; 
but  it  becomes  rectilinear  when  the  direction  V  is  parallel  to  I. 

We  proceed  to  examine  in  some  detail  the  most  important 
cases  of  this  composition  of  two  or  more  simple  harmonic 
motions,  beginning  with  those  cases  in  which  the  resultant 
motion  is  rectilinear. 

As,  according  to  Hooke's  law,  the  particles  of  elastic 
bodies,  after  release  from  strain  within  the  elastic  limits, 
perform  small  oscillations  for  which  the  acceleration  is  pro- 
portional to  the  displacement  from  a  middle  position,  the 
motions  under  discussion  find  a  wide  application  in  the 
theories  of  elasticity,  sound,  light,  and  electricity,  and  form 
the  basis  of  the  general  theory  of  wave  motion  in  an  elastic 
medium. 

77.  Two  simple  harmonic  motions  in  the  same  line,  of  equal 
period  T,  hut  differing  in  amplitude  and  phase,  compound  into 
a  single  simple  harmonic  motion  in  the  same  line  and  of  the 
same  period. 

For,  by  Art.  72,  the  component  displacements  can  be 
written 


60  KINEMATICS  [78. 

Xi  =  ai  COs(co/  +   ei),      Xo  =  02  COs(cof  +   €2), 

and  being  in  the  same  line  they  can  be  added  algebraically, 
giving  the  resultant  displacement 

X  =  Xi  -\-  X2  =  Qi  cos(coi  +   ei)  +  a2  COs(w^  +  eo) 

=  (tti  cosei  +  02  COS62)  coscof  —  (ai  sinei  +  02  sin€2)  sinwi. 

Putting  (comp.  Art.  72) 
ai  costi  +  «2  cose2  =  a  cose,     ai  sinei  +  02  sineo  =  a  sine, 

we  have 

r-- 

X  =  a  cose  coscof  —  sine  smut  =  a  cos(coi  +  e), 
where 

a^  =  (ai  cosei  +  a2  cose2)^  +  (oi  sinei  +  02  sine2)^ 

=  ai^  +  fl2"  +  2aia2  cos(e£  —  ei) 
and 

tti  sine]  +  a2  sine2 


tane  = 


ai  cosei  +  Go  cose2 


78.  A  geometrical  illustration  of  the  preceding  proposition  is  ob- 
tained by  considering  the  uniform  circular  motions  corresponding  to 
the  two  simple  harmonic  motions  (Fig.  22). 


Fig.  22. 

Drawing  the  radii  OPi  =  oi,  OP2  =  (h  so  as  to  include  an  angle 
equal  to  the  difference  of  phase  £2  —  ei  and  completing  the  parallelo- 
gram OP1PP2,  it  appears  from  the  figure  that  the  diagonal  OP  of  this 
parallelogram  represents  the  resulting  amplitude  a. 


80. 


CURVILINEAR   MOTION   OF  A   POINT  61 


As  PiP  is  equal  and  parallel  to  OP 2,  we  have  for  the  projections  on 
any  axis  Ox  the  relation  OPx^,  +  OPx^  =  OP  ,  or  Xi  +  0:2  =  x.  If  now 
the  axis  Ox  be  drawn  so  as  to  make  the  angle  xOPi  equal  to  the  epoch- 
angle  €1,  and  hence  xOPi  =  €2,  the  angle  xOP  represents  the  epoch  e 
of  the  resulting  motion. 

We  thus  have  a  simple  geometrical  construction  for  the  elements  a, 
e  of  the  resulting  motion  from  the  elements  ai,  ei  and  a^,  ez  of  the  com- 
ponent motions.  As  the  period  is  the  same  for  the  two  component 
motions,  the  points  Pi  and  P2  describe  their  respective  circles  with 
equal  angular  velocity  so  that  the  parallelogram  OP1PP2  does  not 
change  its  form  in  the  course  of  the  motion. 

79.  The  construction  given  in  the  preceding  article  can  be  de- 
scribed briefly  by  saying  that  two  simple  harmonic  motions  of  equal 
period  in  the  same  line  are  compounded  by  geometrically  adding  their 
amplitudes,  it  being  understood  that  the  phase-angles  determine  the 
directions  in  which  the  amplitudes  are  to  be  drawn.  Analytically, 
this  appears  of  course  directly  from  the  formulse  of  Art.  77. 

It  follows  at  once  that  not  only  two,  but  any  number  of  simple  har- 
monic motions,  of  equal  period  in  the  same  line,  can  be  compounded  by 
geometric  addition  of  their  amplitudes  into  a  single  simple  harmonic  mo- 
tion in  the  same  line  and  of  the  same  period. 

Conversely,  any  given  simple  harmonic  motion  can  be  resolved  into 
two  or  more  components  in  the  same  line  and  of  the  same  period. 

80.  Exercises. 

(1)  Find  the  resultant  of  three  simple  harmonic  motions  in  the 
same  line,  and  all  of  period  T  =  12  seconds,  the  amplitudes  being  5, 
3,  and  4  cm.,  and  the  phase  dififerences  30°  and  60°,  respectively, 
between  the  first  and  second,  and  the  first  and  third  motions. 

(2)  If  in  the  proposition  of  Art.  77  the  amplitudes  are  equal,  ai  =  02 
=  a,  while  the  phase-angles  differ  by  e2  —  ei  =  5,  show  that  the  re- 
sulting motion  has  the  amplitude  2a  cos|5  and  the  phase-angle  \5: 
(a)  directly,  (6)  from  the  formula;  of  Art.  77,  (c)  by  the  geometric 
method  of  Art.  78. 

(3)  Find  the  resultant  of  two  simple  harmonic;  motions  in  the  same 
line  and  of  equal  period  when  the  amplitudes  are  equal  and  the  phases 
differ:  (a)  by  an  even  multiple  of  w,  (b)  by  an  odd  multiple  of  tt. 

(4)  Resolve  a;  =  10  cos{o)t  +  45°)  info  two  components  in  the  same 


62  KINEMATICS  [81. 

line  with  a  phase  difference  of  30°,  one  of  the  components  having  the 
epoch  0. 

(5)  Trace  the  curves  representing  the  component  motions  as  well  as 
the  resultant  motion  in  Ex.  (1),  taking  the  time  as  abscissa  and  the 
displacement  as  ordinate. 

(6)  Show  that  the  resultant  of  7i  simple  harmonic  motions  of  ec[ual 
period  T  in  the  same  line,  viz. 

Xi  =  Oi  cos  f  *^i  +  «'  ) , 

is  the  isochronous  simple  harmonic  motion 

X  =  o  cos  f  "^  <  4-  e  j, 

where  " 

2^ai  smei 


o^  =  f  ^a;  COSei  )    +  (  YL^^  ^i^^^'  )  '     ^ane 


^aiCOSe; 


81.  The  composition  of  two  or  more  simple  harmonic  mo- 
tions in  the  same  line  can  readily  be  effected,  even  when  the 
components  differ  in  period.  But  the  resultant  motion  is  in 
general  not  simple  har7nonic. 

Thus,  with  two  components 

Xi  =  tti  cos(coi^  +  ei),     X2  =  a2  cos(co2^  +  €2), 

putting  oo2t  -{-  eo  —  coit  -{-  (u2  —  o:i)t  -{-  €2  =  wit  +  ei  -j-  8,  say, 
where  5  =  (co2  —  u)i)t  -\-  eo  —  ei  is  the  difference  of  phase  at 
the  time  t,  we  have  for  the  resulting  motion 

X  =  Xi  -{-  X2  =  ai  cos(a)ii  +  ei)  +  a2  COs(a;i^  +  ei  +  5) ; 
=  (oi  +  02  cos8)  cos(aji^  +  ei)  —  02  sinS  sin(a)ii  +  ei), 

or  putting  ai  +  02  cos5  =  a  cose,     02  sin5  =  a  sine: 

X  =  a  COs(coi^  +  ei  +  e), 
where 

02  sinS 


Oi^  +  a2-  +  2aia2  cos5,     tane 


«!  +  02  cos5' 
5  =  (coo  —  coi)^  -|-  eo  —  ei. 


82.] 


CURVILINEAR   MOTION   OF  A  POINT 


63 


It  can  be  shown  that  this  represents  a  simple  harmonic 
motion  only  when  C02  =  ^  o}\. 

The  formulae  can  be  interpreted  geometrically  by  Fig.  22 
as  in  Art.  78.  But  as  in  the  present  case  the  angle  5,  and 
consequently  the  quantities  a  and  e  in  the  expression  for  x, 
vary  with  the  time,  the  parallelogram  OP1PP2  while  having 
constant  sides  has  variable  angles  and  changes  its  form  in  the 
course  of  the  motion. 


(e)  Wave  motion. 

82.  To  show  the  connection  of  the  present  subject  with 
the  theory  of  wave  motion,  imagine  a  flexible  cord  AB  oi 
which  one  end  B  is  fixed,  while  the  other  A  is  given  a  sudden 


Fig.  23. 


jerk  or  transverse  motion  from  A  to  C  and  back  through  A 
to  D,  etc.  (Fig.  23).  The  displacement  given  to  A  will,  so 
to  speak,  run  along  the  cord,  travelling  from  A  to  B  and 
producing  a  wave,  while  any  particular  point  of  the  cord 


64  KINEMATICS  [83. 

has  approximately  a  rectilinear  motion  at  right  angles  to  AB. 
The  figure  exhibits  the  successive  stages  of  the  motion  up  to 
the  time  when  a  complete  wave  A'K  has  been  produced. 
The  distance  A'K  =  \  is  called  the  length  of  the  wave. 
Let  T  be  the  time  in  which  the  motion  spreads  from  A' to 
K,  that  is,  the  time  of  a  complete  vibration  of  the  point  A, 
from  A  to  C,  back  to  D,  and  back  again  to  A ;  then 

T 

is  called  the  velocity  of  propagation  of  the  wave. 

83.  Suppose  now  that  the  vibration  of  ^  is  a  simple  har- 
monic motion,  say  y  =  a  smcot.  As  the  time  of  vibration 
of  A  is  T  we  must  have  w  =  27r/T',  and  hence 

27r 
CO  =  —  V . 
X 

If  we  assume  that  the  vibrations  of  the  successive  points  of 
the  cord  differ  from  the  motion  of  A  only  in  phase,  the  dis- 
placements of  all  points  of  the  cord  at  any  time  t  can  be 

represented  by 

y  =  a  sm{wt  —  e), 

where  e  varies  from  0  to  27r  as  we  pass  from  A'  to  K. 

If  we  further  assume  that  the  phase-angle  e  of  any  point 
of  the  cord  is  proportional  to  the  distance  x  of  the  point  from 
A'  we  have  e  =  kx,  or  since  e  =  27r  for  x  =  X: 

2t 
e  =  -x. 


Substituting  the  values  of  co  and  e  we  find 

2x 


y  =  a  sm 


^  (Vt  -  x) 


(9) 


The  assumptions  here  made  can  be  regarded  as  roughly 


85.]  CURVILINEAR   MOTION   OF   A   POINT  65 

suggested  by  the  experiment  of  Art.  82  or  similar  observa- 
tions. The  motion  represented  by  the  final  equation  (9)  may 
be  called  simple  harmonic  wave  motion. 

84.  To  understand  the  full  meaning  of  the  equation  (9)  it 
should  be  observed  that,  as  (in  accordance  with  the  assump- 
tions of  Art.  83)  the  quantities  a,  X,  V  are  regarded  as 
constant,  the  displacement  ?/  is  a  function  of  the  two  variables 
t  and  X. 

If  t  be  given  a  particular  value  h,  equation  (9)  represents 
the  displacements  of  all  points  of  the  cord  at  the  time  h. 
The  substitution  for  x  oi  x -\-  n\,  where  n  is  any  positive  or 
negative  integer,  changes  the  angle  (2x/X)  {Vt  —  x)  by  2x71 
and  hence  leaves  y  unchanged.  This  means  that  the  dis- 
placements of  all  points  whose  distances  from  A  differ  by 
whole  wave-lengths  are  the  same;  in  other  words,  the  state 
of  motion  at  any  instant  is  given  by  a  series  of  equal  waves. 

If,  on  the  other  hand,  we  assign  a  particular  value  Xi  to  x 
and  let  t  vary,  the  equation  represents  the  rectilinear  vibra- 
tion of  the  point  whose  abscissa  is  Xi.  By  substituting  for  t 
the  value  t  +  nT  =  t -^  nK/V,  the  angle  (27r/X)(F^  -  x)  is 
again  changed  by  2Tn,  so  that  y  remains  unchanged.  This 
shows  the  periodicity  of  the  motion  of  any  point. 

85.  It  may  be  well  to  state  once  more,  and  as  briefly  as  possible, 
the  fundamental  assumptions  that  underlie  the  important  formula  (9); 

The  idea  of  simple  harmonic  wave  motion  implies  that  the  dis- 
placement y  should  be  a  periodic  function  of  x  and  t  such  as  to  fulfil 
the  following  conditions:  y  must  assume  the  same  value  (a)  when  x 
is  changed  to  x  +  nX,  (h)  when  t  is  changed  to  t  +  nT,  (c)  when  both 
changes  are  made  simultaneously;  the  constants  X  and  T  being  con- 
nected by  the  relation  X  =  VT. 

The  condition  {c)  rociuires  y  to  be  of  the  form  y  =  f{Vt  —  x);  for 
Vt  —  X  remains  unchanged  when  x  is  replaced  by  x  +  nX  and  at  the 
same  time  ^  by  ^  +  nT. 
6 


66  KINEMATICS  [86. 

A  particular  case  of  such  a  function  is  y  =  a  sinc(T'/  —  x).  As 
y  should  remain  unchanged  when  t  is  replaced  by  I  +  T,  we  must 
have  c  =  2ir/FT  =  27r/X.     Thus  the  function 

y  =  a  sin  "";^    {Vt  —  x) 

fulfils  the  three  conditions  (a),  (b),  (c). 
Putting  2irxJ\  =  —  6  we  have 


y  =  a  sinl'^^  t  +eY 


The  importance  of  this  particular  solution  of  our  problem  lies  in 
the  fact  that,  according  to  Fourier's  theorem,  any  single-valued  periodic 
function  of  period  T  can  be  expanded,  between  definite  limits  of  the 
variable,  in  a  series  of  the  form: 

fit)  =  ao  +  ai  sin  (-^-t  +  eij  +  02  sin  f  ^    ■  2/  -|-  €2  j 

+  Ui  sin  (-1,  ■  Zt  +  i3  )  +   ■■  ■ . 

As  applied  to  the  theory  of  wave  motion  this  means  that  any  wave 
motion,  however  complex,  can  be  regarded  as  made  up  of  a  series  of 
superposed  simple  harmonic  wave  motions  of  periods  T,  \T,  \T,  .  .  ., 
or  since  T  =  X/V,  of  wave-lengths  X,  IX,  |X,  ....  For,  if  the  point 
A  (Fig.  23)  be  subjected  simultaneously  to  more  than  one  simple 
harmonic  motion,  the  displacements  resulting  from  each  can  be  added 
algebraically,  thus  forming  a  compound  wave  which  can  readily  be 
traced  by  first  tracing  the  component  waves  and  then  adding  their 
ordinates. 

The  motion  due  to  the  superposition  of  two  or  more  simple  harmonic 
waves  may  be  called  compound  harmonic  irave  motion. 

86.  Exercises. 

(1)  Trace  the  wave  produced  by  the  superposition  of  two  simple 
harmonic  wave  motions  in  the  same  line  of  equal  amplitudes,  the 
periods  being  as  2  :  1,  (a)  when  they  do  not  differ  in  phase,  (6)  when 
their  epochs  differ  bj^  7/16  of  the  period. 

(2)  In  the  problem  of  Art.  81,  determine  the  maximum  and  mini- 
mum of  the  resulting  amplitude  a  and  show  that  the  number  of  maxima 


87.] 


CURVILINEAR   MOTION  OF  A   POINT 


67 


per  second  is  equal  to  the  difference  of  the  number  of  vibrations  per 
second. 

(f)  Curvilinear  compound  harmonic  motion. 

87.  An  important  and  typical  case  is  the  motion  of  a  point 
P  whose  acceleration  j  is  directed  toward  a  fixed  center  0 


y 

!\ 

^ 

Py 

^^ 

VP 

^ 

Tj^ 

J 

i> 

0 

X 

Px 

Fig.  24. 

and  proportional  to  the  distance  OF  =  r  from  this  center 
(Fig.  24). 

If  the  initial  velocity  is  =1=  0  and  does  not  happen  to  pass 
through  the  center  0,  the  motion  is  curvilinear.  But  it  is 
confined  to  the  plane  determined  by  the  center  and  the 
initial  velocity  since  the  acceleration  j  =  yih  hcs  in  this  plane. 

Taking  the  center  0  as  origin  and  any  rectangular  axes 
Ox,  Oy  in  this  plane,  we  have  for  the  direction  cosines  of  OP: 
x/r,  y/r,  for  those  of  the  acceleration :  —  x/r,  —  y/r,  so  that 
the  equations  of  motion  are 


x  = 


n'x,  y 


fj^^y- 


These  equations  show  that  the  projections  Px,  Py  of  P  on 
the  axes  have  each  a  simple  harmonic  motion,  of  the  same 
center  and  period.  The  motion  of  P  is  the  absolute  motion 
of  a  point  having  a  simple  harmonic  motion  of  period  27r/jLt 


68  KINEMATICS  [88. 

along  the  axis  Ox,  about  0,  while  this  axis  itself  has  a  simple 
harmonic  motion  of  the  same  period  about  0  along  the  axis 
Oy. 

Each  of  the  two  equations  is  readily  integrated,  and  by 
eliminating  t  it  is  found  that  the  path  is  an  ellipse,  with  0 
as  center.     See  Arts.  298-302. 

88.  To  corn-pound  any  number  of  simple  harmonic  motions  not  in  the 
same  line  observe  that  the  projection  of  a  simple  harmonic  motion  on 
any  hne  is  again  a  simple  harmonic  motion  of  the  same  period  and 
phase  and  with  an  amplitude  equal  to  the  projection  of  the  original 
amplitude. 

For  the  sake  of  simplicity  we  confine  ourselves  to  the  case  of  motions 
in  the  same  plane  and  with  the  same  center  0.  Projecting  all  the 
simple  harmonic  motions  on  two  rectangular  axes  Ox,  Oy,  we  can,  by 
Arts.  77,  79,  compound  the  components  in  each  axis;  it  then  only  re- 
mains to  find  the  resultant  of  the  two  motions  along  Ox  and  Oy. 

89.  Just  as  in  Arts.  77,  81,  we  must  distinguish  two  cases:  (o)  When 
the  given  motions  have  all  the  same  period,  and  (6)  when  they  have  not. 

In  the  former  case,  by  Art.  77,  the  two  components  along  Ox  and 
Oy  will  have  equal  periods,  i.  e.  they  will  be  of  the  form 

X  =  a  coscot,     y  =  b  cos(wt  +  5). 

The  path  of  the  resulting  motion  is  obtained  by  eliminating  t  be- 
tween these  equations.     We  have 


cosut  cos5  —  sinut  sin6 


^-cos5-  Jl-^^ 
a  \  a^ 


Writing  this  equation  in  the  form 


-?-  C085  +  f,  =  sin^S,  (10) 

ab  b^ 


11        cos^S        /  sinS  \2 
see  that  it  represents  an  ellipse  (since  ^  ^  '  1,2  ~     2i,f  ~  \      fT  ) 


91.]  CURVILINEAR  MOTION   OF   A   POINT  69 

is  positive)  whose  center  is  at  the  origin.  The  resultant  motion  is 
therefore  called  elliptic  harmonic  motion. 

We  have  thus  the  general  result  that  any  number  of  simple  harmonic 
motions  of  the  same  -period  and  in  the  same  plane,  whatever  may  be  their 
directions,  amplitudes,  and  phases,  cornpoiutd  into  a  single  elliptic  har- 
monic motion. 

90.  A  few  particular  cases  may  be  noticed.  The  equation  (10)  will 
represent  a  (double)  straight  line,  and  hence  the  elliptic  vibration  will 
degenerate  into  a  simple  harmonic  vibration,  whenever  sin^S  =  0,  i.  e. 
when  8  =  nir,  where  n  is  a  positive  or  negative  integer.  In  this  case 
cos5  is  +  1  or  —1,  and  (10)  reduces  to 


and  to 


'^  =  0,  if  5  =  27mr, 


-  +  ■'   =  0,  if  5  =  (2/«  +  1)^. 
a        b 


Thus  two  rectangular  vibrations  of  the  same  period  compound  into 
a  simple  harmonic  vibration  when  they  differ  in  phase  by  an  integral 
multiple  of  it,  that  is  when  one  lags  behind  the  other  by  half  a  wave- 
length. 

Again,  the  ellipse  (10)  reduces  to  a  circle  only  when  cosS  =  0,  i.  e. 
6  =  {2m.  +  l)7r/2,  and  in  addition  a  =  b,  the  co-ordinates  being  as- 
sumed orthogonal. 

Thus  two  rectangular  vibrations  of  equal  period  and  amplitude  com- 
pound into  a  circular  vibration  if  they  differ  in  phase  by  7r/2,  i.  e.  if 
one  lags  behind  the  other  by  a  quarter  of  a  wave-length. 

This  circular  harmonic  motion  is  evidently  nothing  but  uniform 
motion  in  a  circle;  and  we  have  seen  in  Art.  73  that,  conversely,  uniform 
circular  motion  can  be  resolved  into  two  rectangular  simple  harmonic 
vibrations  of  equal  period  and  amplitude,  but  differing  in  phase  by  ir/2. 

91.  It  remains  to  consider  the  case  when  the  given  simple  harmonic 
motions  do  not  all  have  the  same  period.  It  follows  from  Art.  81 
that  in  this  case,  if  we  again  project  the  given  motions  on  two  rec- 
tangular axes  Ox,  Oy,  the  resulting  motions  along  Ox,  Oy  are  in  general 
not  simple  harmonic. 

The  elimination  of  t  between  the  expressions  for  x  and  y  may  present 
difficulties.  But,  of  course,  the  curve  can  always  be  traced  by  points, 
graphically. 


70  KINEMATICS  [92. 

We  shall  here  consider  only  the  case  when  the  motions  along  Ox 
and  Oy  are  simple  harmonic. 

92.  If  two  simple  harmonic  motions  along  the  rectangular  directions 
Ox,  Oy,  viz.: 

X  =  ai  cos(wi<  +  ei),     y  =  02  cos(co2<  +  €2), 

of  different  amplitudes,  phases,  and  periods  are  to  be  compounded,  the 
resulting  motion  will  be  confined  within  a  rectangle  whose  sides  are 
2ai,  2a;,  since  these  are  the  maximum  values  of  2x  and  2y. 

The  path  of  the  moving  point  will  be  a  closed  curve  only  when  the 
quotient  C02/C01  =  Ti/T-,  is  a  rational  number,  say  =  7n/?i,  where  m 
is  prime  to  n.  The  x  co-ordinate  of  the  curve  will  have  m  maxima,  the 
y  co-ordinate  n,  and  the  whole  curve  will  be  traversed  after  m  vibrations 
along  Ox  and  n  along  Oy. 

The  formation  of  the  resulting  curve  will  best  be  understood  from 
the  following  example. 

93.  Let  fli  =  a2  =  a,  «i  =  0,  €2  =  5,  and  let  the  ratio  of  the  periods 

be  T1IT2  =  2/1.     The  equations  of  the  component  simple  harmonic 

vibrations  are 

X  =  a  coswt,     y  =  a  cos(2a)i  -\-  8). 

Here  it  is  easy  to  eliminate  t.     We  have 

y  =  a  cos2co<  cos5  —  a  sm2wt  sinS 

2 ' ,  —  1  )  cos5  —  2a  -  \1  —  - ,  sinS. 
a^         J  a  \         a' 

Hence  the  equation  of  the  path  is: 


ay  =  (2z'  —  a^)  cos5  —  2x  Va-  —  x^  sinS. 

If  there  be  no  difference  of  phase  between  the  components,  i.  e.  if 
5=0,  this  reduces  to  the  equation  of  a  parabola: 

x^  =  ia(y  -f-  a). 

For  5  =  ir/2,  the  equation  also  assumes  a  simple  form: 

(j2y2    —    4^2  (a2    _    2;2)_ 

94.  It  is  instructive  to  trace  the  resulting  curves  for  a  given  ratio 
of  periods  and  for  a  series  of  successive  differences  of  phase  {Lissajous's 
Curves) . 

Thus  in  Fig.  25,  the  curve  for  TJTo  =  H,  and  for  a  phase  differ- 
ence 5  =  0  is  the  heavily  drawn  curve,  while  the  dotted  curve  repre- 


93. 


CURVILINEAR   MOTION   OF  A   POINT 


71 


sents  the  path  for  the  same  ratio  of  the  periods  when  the  phase  difference 
is  one-twelfth  of  the  smaller  period.  The  equations  of  the  components 
are  for  the  heavy  cm-ve 

X  =  D  cos       /,     y  =  o  cos ~~.~ t, 
and  for  the  dotted  curve 

„  /  2x  27r  V  2x 

X  =  6  cos  (  ^  /  +  ^.^  1  ,     2/  =  5  cos  —  /. 

In  tracing  these  curves,  imagine  the  simple  harmonic  motions  re- 
placed   by  the   corresponding    uniform   circular    motions    (Fig.    25). 


E^ 

/ 

y 

^ 

^ 

.^--' 

---,,^ 

^::>^;:^ 

>< 

/ 

\ 

'"^•v,^ 

^^' 

^>< 

X 

i 

\ 

JSvI 

^xT^ 

0.-^" 

\ 

X 

X 

I 

/ 

><^ 

"\. 

y\ 

i 

\ 

/ 

,.'•'' 

'""*^,^ 

^x^ 

X 

/ 

X, 

~^---, 

^>-<r^ 

s< 

\^ 

A 

V 

/ 

0 

B 

Fig.  25. 

With  the  amplitudes  6,  5,  as  radii,  describe  the  semi-circles  ADB, 
AEC,  so  that  BC  is  the  rectangle  within  which  the  curves  are  con- 
fined; the  intersection  of  the  diagonals  of  this  rectangle  is  the  origin 
0,  AB  is  parallel  to  the  axis  of  x,  AC  to  the  axis  of  y.  Next  divide 
the  circles  on  AB,  AC  into  parts  corresponding  to  equal  intervals 
of  time.  In  the  present  case,  the  periods  for  AB,  AC  being  as  3  to 
4,  the  circle  on  AB  must  be  divided  into  3n  equal  parts,  that  on  AC 
into  4n.  In  the  figure,  n  is  taken  as  4,  the  circles  being  divided  into 
12  and  16  equal  parts,  respectively. 


72       -  KINEMATICS  [95. 

The  first  point  of  the  heavily  drawn  curve  corresponds  to  <  =  0, 
that  is  X  =  Q,  y  =  5;  this  gives  the  upper  right-hand  corner  of  the 
rectangle.  The  next  point  is  the  intersection  of  the  vertical  line  through 
D  and  the  horizontal  line  through  E,  the  arcs  BD  =  1/12  of  the  circle 
over  AB,  and  CE  =  1/16  of  that  over  AC  being  described  in  the  same 
time,  so  that  the  co-ordinates  of  the  corresponding  point  are 

a;  =  6  cos  (^  y  '  12  )  "  ^  ^^^  V"" '  1 2) ' 

2/  =  5co3("f:.^)=5eos(2..^). 
Similarly  the  next  point 

X  =  6  cos  f  27r  •  ^  j ,     ?/  =  5  cos  f  2jr  •  y'Jv  j 

is  found  from  the  next  two  points  of  division  on  the  circles,  etc. 

To  construct  the  dotted  curve,  it  is  only  necessary  to  begin  on  the 
circle  over  AB  with  D  as  first  point  of  division, 

95.  Exercises. 

(1)  With  the  data  of  Art.  94  construct  the  curves  for  phase  differ  • 
ences  of  2/12,  3/12,  .  .  .  11/12  of  the  smaller  period. 

(2)  Construct  the  curves  (Art.  93) 

X  =  a  coscoi,     y  =  a  cos(2cof  -|-  5) 

for  5=0,  7r/4,  7r/2,  37r/4,  TT,  Bx/i,  3w/2,  77r/4,  27r. 

(3)  Trace  the  path  of  a  point  subjected  to  two  circular  vibrations 
of  the  same  amplitude,  but  differing  in  period:  (a)  when  the  sense  is 
the  same;  (b)  when  it  is  opposite. 

(g)  Central  motion. 

96.  The  motion  of  a  point  P  is  called  central  if  the  direction 
of  the  acceleration  always  passes  through  a  fixed  point  0. 
It  will  here  be  assumed,  in  addition,  that  the  magnitude  of 
the  acceleration  is  a  function  of  the  distance  OP  =  r  alone,  say 

i  =  f(r). 
The  fixed  point  0  is  in  this  case  usually  regarded  as  the 
seat  of  an  attractive  or  repulsive  force  producing  the  accelera- 
tion, and  is  therefore  called  the  center  of  force. 


98.1  CURVILINEAR   MOTION   OF   A   POINT  73 

Harmonic  motion  as  discussed  in  Art.  71  sq.  is  a  special 
case  of  central  motion,  viz.  the  case  in  which  the  acceleration 
j  is  directly  proportional  to  the  distance  from  the  fixed  center 
0,  i.  e.  J{r)  =  )Lir;  see  Art.  87. 

Another  very  important  particular  case  is  that  of  Newton's 
law,  i.  e.  f(r)  —  n/r-;  this  will  be  discussed  below,  Arts. 
109-112. 

97.  Any  central  motion  is  fully  determined  if  in  addition 
to  the  form  of  the  function  f(r)  we  know  the  "  initial  condi- 
tions," say  the  initial  distance  OPo  =  ro  (Fig.  26)  and  the 


Fig.  26. 

initial  velocity  Vo  of  the  point  at  the  time  ^  =  0.  As  Vo  must 
be  given  both  in  magnitude  and  direction,  the  angle  \po  be- 
tween ro  and  vo  must  be  known. 

It  is  evident,  geometrically,  that  the  motion  is  confined 
to  the  plane  determined  by  0  and  Vo  since  the  acceleration 
always  lies  in  this  plane.  Hence,  any  central  motion,  what- 
ever may  be  the  law  of  acceleration,  is  a  plane  motion. 

98.  Another  fundamental  property  is  that  in  any  central 
motion,  whatever  the  law  of  acceleration,  the  sectorial  velocity 
is  constant.  This  is  most  readily  proved  by  taking  the  center 
0  as  origin  for  polar  co-ordinates  r,  d.  As  by  the  definition 
of  central  motion  (Art.  96)   the  acceleration  j  is  directed 


74  KINEMATICS  [99. 

along  the  radius  vector  OP  =  r  drawn  from  the  center  0  to 
the  moving  point  P,  the  component  je  of  the  acceleration, 
at  right  angles  to  the  radius  vector,  is  always  zero.  We  have 
therefore  by  the  l9.st  of  the  equations  of  Art.  55 : 

whence  -    -X 

'   r'f  =  ^.  (11) 

where  c  is  the  constant  of  integration.  By  Art.  47  this 
equation  means  that  the  sectorial  velocity  is  constant  and 
equal  to  ic. 

99.  Let  S  be  the  sector  PoOP  described  by  the  radius 
vector  r  in  the  time  t,  so  that  dS  =  h'-dd  is  the  elementary 
sector  described  in  the  element  of  time  dt.     Then  (11)  can 

be  written 

dS  _  1 
_  dt   ~  ''' 

whence  integrating,  since  S  =  0  ior  t  =  0: 

S  =  id. 

This  shows  that  the  sector  is  proportional  to  the  time  in  which 
it  is  described,  which  is  merely  another  way  of  stating  that  the 
sectorial  velocity  is  constant. 

It  can  be  shown  conversely,  by  reversing  the  steps  of  the 
above  argument,  that  if  in  a  plane  motion  the  areas  swept 
out  by  the  radius  vector  drawn  from  a  fixed  point  of  the  plane 
are  proportional  to  the  time,  the  acceleration  must  constantly 
pass  through  that  point 

It  is  well  known  that  Kepler  had  found  by  a  careful  exami- 
nation of  the  observations  available  to  him  that  the  orbits 
described  by  the  planets  are  plane  curves,  and  the  sector  described 


101.1  CURVILINEAR  MOTION  OF  A  POINT  75 

by  the  radius  vector  drawn  from  the  sun  to  any  planet  is  pro- 
portional to  the  time  in  which  it  is  described.  This  constitutes 
Kepler's  first  law  of  planetary  motion. 

He  concluded  from  it  that  the  acceleration  must  constantly- 
pass  through  the  sun. 

100.  To  express  the  value  of  the  constant  of  integration  c 
in  terms  of  the  given  initial  conditions  (Art.  97),  i.  e.  by 
means  of  ro,  Vo,  \l/o,  observe  that  at  any  time  t 

^dd  rdd    ds  .    , 

c  =  r^-r-  =  r-  -T-  •  TT  —  f  smi/"?;; 
dt  ds    at 

hence  at  the  time  t  =  0  we  have 

c  =  voro  sini/'o. 
Denoting  l^y  po  and  p  the  perpendiculars  let  fall  from  0 
on  Vo  and  v  we  have  ro  sim/^o  =  pa,  r  sinij/  =  p;  hence 

c  =  poVo  =  pv, 

i.  e.  the  velocity  at  any  tim£  is  inversely  proportional  to  its 
distance  from  the  center. 

101.  Let  us  now  assume  that  the  acceleration  j  of  a  central 
motion  is  a  given  function,  f(r),  of  the  radius  vector  OP  =  r 
drawn  from  the  center  0  to  the  moving  point  P.  With  0 
as  origin,  let  x,  y  be  the  rectangular  cartesian  co-ordinates 
of  the  moving  point  P,  and  r,  d  its  polar  co-ordinates,  at  any 
time  t.  Then  cos0  =  xjr,  m\Q  =  y/r  are  the  direction  cosines 
of  OP  =  r,  and,  therefore,  those  of  the  acceleration  j,  pro- 
vided the  sense  of  j  be  away  from  the  center,  i.  e.  provided  the 
force  causing  the  acceleration  be  repulsive.  In  the  case  of 
attraction,  the  direction  cosines  of  j  are  of  course  —  x/r,  —  y/r. 

Thus  the  equations  of  motion  are  in  the  case  of  attraction : 

x=-f(r)f,      y=^-f{r)f.  (12) 


76  KINEMATICS  [i02. 

For  repulsion,  it  would  only  be  necessary  to  change  the 
sign  of /(rj. 

To  integrate  the  equations  (12)  we  cannot,  in  general,  treat 
each  equation  by  itself;  for,  as  r  ==  '\x^  +  y'^,  each  equation 
contains  three  variables  x,  y,  t.  We  must  therefore  try  to 
combine  the  equations  so  as  to  form  integrable  combinations. 

102.  Let  us  first  multiply  the  equations  (12)  by  y,  x  and 
subtract;  the  right-hand  member  of  the  resulting  equation  is 
zero,  while  the  left-hand  member  is  an  exact  derivative: 

xy  -  yx  ^-^{xy  -  yx)  =  0. 

Integrating  we  find  xy  —  yx  =  c,  or  in  polar  co-ordinates 

r^'d  =  c, 

which  is  the  equation  (11)  of  Art.  98. 

103.  Next  multiply  the  equations  (12)  by  x,  y  and  add; 
the  left-hand  member  of  the  resulting  equation  is 

ix  +  yy  =  ^a^-^m=^h^; 

the  right-hand  member  becomes 

-i^ixi  +  yy)  =  -^-y-^Ux'  +  y')  - 

The  resulting  equation 

div^  =  -  f{r)dr  (13) 

gives 

^2  _  ^,^2  =  _   rf{r)dr',  (14) 

i.  e.  it  determines  the  velocity  as  a  function  of  r. 


105.1 


CURVILINEAR  MOTION   OF  A  POINT 


77 


104.  The  two  methods  of  combining  the  differential  equa- 
tions of  motion  (12)  used  in  Arts.  102  and  103  are  known, 
respectively,  as  the  principle  of  areas  and  the  jjrinciple  of 
kinetic  energy  and  work.  The  former  name  explains  itself 
(see  Arts.  98,  99).  The  latter  is  due  to  the  fact  (to  be  more 
fully  explained  in  kinetics)  that  if  equation  (14)  be  multiplied 
liy  the  mass  of  the  moving  body,  the  left-hand  member  will 
represent  the  increase  in  kinetic  energy  while  the  right-hand 
member  is  the  work  of  the  central  force. 

Each  of  these  methods  of  preparing  the  equations  of  motion 
for  integration  consists  merely  in  combining  the  equations  so 
as  to  obtain  an  exact  derivative  in  the  left-hand  member  of 
the  resulting  equation.  If  by  this  combination  the  rights 
hand  member  happens  to  vanish  or  to  become  likewise  an 
exact  derivative,  an  integration  can  at  once  be  performed. 
This  is  the  case  in  our  problem. 

105.  The  two  equations  (11)  and  (14),  each  of  which  was 
found  by  a  first  integration,  are  called  first  integrals  of  the 
equations  of  motion.  By  combining  them  and  integrating 
again,  the  equation  of  the  path  is  found. 

We  have,  by  the  last  equation  of  Art.  46,  for  any  curvilinear 
motion 

eliminating  dd/dt  by  means  of  (11)  we  find  for  any  central 
motion: 


\dtl  ^      \dt 


drV  ,     , 

de)-^'~ 


+  r= 


ddr 


+ 


(15) 


Substituting  this  expression  of  v-  in  (14)  we  have  the  dif- 
ferential equation  of  the  path  in  which  the  variables  are 
separable.  Shorter  methods  may  occasionally  suggest  them- 
selves in  particular  cases;  see,  for  instance,  Art.  110. 


78  KINEMATICS  [106. 

106.  To  solve  the  converse  problem,  viz.  to  find  the  law  of 
acceleration  when  the  path  is  known,  we  have  only  to  sub- 
stitute the  expression  (15)  of  v'^  in  the  equation  (13). 

In  doing  this  it  is  found  convenient  to  introduce  instead 
of  r  its  reciprocal  u  —  1/r,  so  that 


{fsr-4 


and  to  change  the  r-derivative  of  h>'  to  a  0-derivative  since 
r,  and  hence  u,  is  now  a  given  function  of  6.     As  du/dr  = 

..  .  _        c^  J.  2  _       ^^^"  ^^^  —       4j1^  ^^  ^^ 
•'^^^  ~  ~  dr^^'    ~  ~  ~dd  dr~  ~    dd  dudr 

_     ,^dd   d  ^  ^  _    ^  ^dd  / du  d'hi         du 

^  ""  dadd'^'^  ~  ^'''*  dM\dddd'^  ""dd- 


hence 


Kr)  =  c^w^(^^  +  t.).  (16) 


This  important  relation  can  also  be  obtained  directly  from 
the  equations  of  motion  in  polar  co-ordinates  which  are 
(see  Art.  55) 

f  -  rd^  =  -  f{r),     ^  4r(r4)  =  0. 

For,  with  r  —  1/w  we  have  since  the  second  equation  gives 
(11): 

dr  ■       c  dr  du        ..  dhi  :  .     Su 

'  =  de^^r^de^-'W     '=-'de'^^  =  -'''de-^' 

substituting  these  values  in  the  first   equation  we  find  (16). 
107.  Kepler  in  his  second  law  had  cstal)lished  the  empirical 
fact  that  the  orbits  of  the  planets  are  ellipses,  with  the  sun  at 
one  of  the  foci. 


108.]  CURVILINEAR  MOTION  OF  A  POINT  79 

From  this  Newton  concluded  that  the  law  of  acceleration 
must  be  that  of  the  inverse  square  of  the  distance  from  the 
sun.  Our  equation  (16)  enables  us  to  draw  this  conclusion. 
The  polar  equation  of  an  ellipse  referred  to  focus  and  major 
axis  is 

r  = -— -,      I.e.      u=Y  +  rCosd, 

1  +  e  cos^  I       I 

where  I  =  h^/a  =  a(l  —  e^);  a,  b  being  the  semi-axes,  I  the 
semi-latus  rectum,  and  e  the  eccentricity.     Hence 

d~u  e        „      dhi  ,  1  . 


so  that  we  find 


p2,  pi  1 


108.  The  third  law  of  Kepler,  found  by  him  likewise  as  an 
empirical  fact,  asserts  that  the  squares  of  the  periodic  times  of 
different  -planets  are  as  the  cubes  of  the  major  axes  of  Iheir  orbits. 

From  this  fact  Newton  drew  the  conclusion  that  in  the 
law  of  acceleration, 

the  constant  n  has  the  same  value  for  all  the  planets. 

Our  formulse  show  this  as  follows.  Let  T  be  the  periodic 
time  of  any  planet,  i.  e.  the  time  of  describing  an  ellipse  whose 
semi-axes  are  a,  b.  Then,  since  the  sector  described  in  the 
time  T  is  the  area  irab  of  the  whole  ellipse,  we  have  by  Art.  99 

-wab  =  icT. 

Substituting  in  (17)  the  value  of  c  found  from  this  equation 
we  have 


80  KINEMATICS  [109. 

Hence 


is  constant  by  Kepler's  third  law, 

109.  Planetary  motion  in  its  simplest  form  is  that  particular  case 
of  aentral  motion  in  which  the  acceleration  is  inversely  proportional 
to  the  square  of  the  distance  from  the  center  0  so  that 

where  m  is  a  constant,  viz.,  the  acceleration  at  the  distance  r  =  1 
from  0. 

The  equations  of  motion  (12)  are  in  this  case,  with  O  as  origin, 

df=  -  ^r^'     d^^  ~''r'y  (^^) 

Combining  these  by  the  principle  of  energy  (Arts.  103,  104),  we  find 

Integrating  the  differential  equation  d^v-  =  —  (jj.lr'^)dr  we  find 

lj,2_^j,.2=if  _if  .  (19) 

r       ro 

110.  To  find  the  equation  of  the  path,  or  orbit,  write  the  equations 

(IS)  in  the  form 

X  =  —  -„  cos9,     ij  =  —    „  sin9 
r-  r- 

and  eliminate  r^  by  means  of  (11): 

X  =  -  -  cose  -e,     y  =  -  -  sine  •  d. 
c  c 

Each  of  these  equations  can  be  integrated  by  itself: 

X  -V,  =  -  :^  sine,     ij  -  V,  =^  (cose  -  1),  (20) 

where  vi,  V2  are  the  components  of  the  velocity  when  e  =  0. 


112.]  CURVILINEAR  MOTION  OF  A  POINT  81 

Multiplying  by  y,  x,  subtracting,  and  integrating,  we  find  by  Art. 
102: 

i~-Vi\x  +  Viy-\-c=-~{x  cose  +  y  amd)  =  ~  Vx^  +  y"".     (21) 

111.  The  geometrical  meaning  of  this  equation  is  that  the  radius 
vector  r  =  Vx"^  +  y~  drawn  from  the  fixed  point  0  to  the  moving 
point  P  is  proportional  to  the  distance  of  P  from  the  fixed  straight  line 

{j-^  -  v^Yx  +  viy  +  c  =  Q.  (22) 

It  represents,  therefore,  a  conic  section  having  O  for  a  focus  and  the 
line  (22)  for  the  corresponding  directrix. 

The  character  of  the  conic  depends  on  the  absolute  value  of  the 
ratio  of  the  radius  vector  to  the  distance  from  the  directrix;  according 
as  this  ratio 


.S'(^ --)'+».'. 


is  <  1.  =  1,  or  >  1,  the  conic  will  be  an  ellpise,  a  parabola,  or  a  hyper- 
bola. This  criterion  can  be  simplified.  Multiplying  by  njc  and 
squaring,  we  have 

or  since  v^  +  vn}  —  vi  and  c  =  nvo  sin>/'o  =  rav^: 

t;o2^-''.1  (23) 

112^  If  polar  co-ordinates  be  introduced  in  (21),   the  equation  of 
the  orbit  assumes  the  form 

1  =M  +  fL^_ii')cos0--'6in5, 

T  &         \  C  C-'  )  C 

or  putting  {cv^  —  ii)lc^  =  C  cosa,  th/c  =  C  sina, 

1  =  -^  +  C  cos(9  +  a).  (24) 

r      c^ 

This  equation  might  have  been  obtained  dhectly  by  integrating 
(16),  which  in  our  case,  with/(r-)  =  iJi/r-,  reduces  to 

^^   1    ,    1  ^  M  . 
dm  r^  r      c^  ' 


82  KINEMATICS  (113. 

the  general  integral  of  this  differential  equation  is  of  the  focm  (24), 
C  and  a  being  the  constants  of  integration. 

Equation  (24)  represents  a  conic  section  referred  to  the  focus  as 
origin  and  a  line  making  an  angle  a.  with  the  focal  axis  as  polar  axis. 

113.  Exercises. 

(1)  A  point  moves  in  a  circle;  if  the  acceleration  be  constant  in  direc- 
tion, what  is  its  magnitude? 

(2)  A  point  describes  a  circle;  if  the  acceleration  be  constantly 
directed  towards  the  center,  what  is  its  magnitude? 

(3)  A  point  has  a  central  acceleration  proportional  to  the  distance 
from  the  center  and  directed  away  from  the  center;  find  the  equation 
of  the  path. 

(4)  A  point  P  is  subject  to  two  accelerations,  ^^  -  0\P  directed 
toward  the  fixed  point  0\,  and  ^t^  o^P  directed  awaj^  from  the  fixed 
point  O2.     Show  that  its  path  is  a  parabola. 

(5)  A  point  P  describes  an  ellipse  owing  to  a  central  acceleration 
^('■)  =  i"/^^  directed  toward  the  focus  S.  Its  initial  velocity  i^o  makes 
an  angle  yp^  with  the  initial  radius  vector  Td.  Determine  the  semi-axes 
a,  h  of  the  ellipse  in  magnitude  and  position. 

(6)  Find  the  law  of  acceleration  when  the  equation  of  the  orbit  is 
j.n  =  5''/(l  4-  e  cosnO),  e  being  positive,  and  investigate  the  particular 
cases  n  =  I,  n  =  2,  n  =  —  1,  71  =  —  2. 

(7)  Find  the  law  of  the  central  acceleration  directed  to  the  origin 
under  whose  action  a  point  will  describe  the  following  curves:  (a)  the 
spiral  of  Archimedes  r  =  ad;  (6)  the  hyperbolic  spiral  Or  =  a;  (c)  the 
logarithmic  or  equiangular  spiral  r  =  ae"^;   (d)  the  curve  r  =  a  cosnd. 

(8)  A  point  moves  in  a  circle  and  has  its  acceleration  directed 
towards  a  point  on  the  circumference.     Find  the  law  of  acceleration. 

(9)  The  acceleration  of  a  point  is  perpendicular  to  a  given  plane  and 
inversely  proportional  to  the  cube  of  the  distance  from  the  plane. 
Determine  its  motion. 

(10)  A  point  moves  in  a  semi-ellipse  with  an  acceleration  perpen- 
dicular to  the  axis  joining  the  ends  of  the  semi-ellipse.  Determine 
the  law  of  acceleration  and  the  velocity. 


CHAPTER  IV. 
VELOCITIES  IN  THE  RIGID  BODY. 

1.  Geometrical  discussion. 

114.  The  velocities  of  the  various  points  of  a  rigid  body, 
at  any  instant,  are  in  general  different,  both  in  magnitude 
and  direction;  i.  e.  they  are  different  vectors;  but  they  are 
not  independent  of  each  other 

In  particular,  it  is  clearly  possible  (i.  e.  compatible  with 
the  rigidity  of  the  body)  that  the  velocities  of  all  points,  at 
the  given  instant,  are  equal  vectors.  The  instantaneous 
state  of  motion  of  the  body  is  then  called  a  translation;  it  is 
fully  determined  by  the  velocity  vector  of  an}^  one  point  of 
the  body,  and  this  is  called  the  velocity  of  translatio7i,  or 
linear  velocity,  of  the  body.     Comp.  Art.  30. 

The  ideas  of  absolute  and  relative  velocity  and  of  composi- 
tion and  resolution  of  velocities  apply  to  the  velocity  of  trans- 
lation of  a  rigid  l^ody  just  as  they  apply  to  the  linear  velocity 
of  a  point  (comp.  Arts.  38,  40,  41). 

115.  As  the  position  of  a  rigid  body  is  fully  determined  by 
the  positions  of  any  three  of  its  points,  Oi,  O2,  O3,  not  in  a 
straight  line,  it  is  clear  that  if  any  three  such  points  have 
zero  velocity  at  any  instant,  all  points  of  the  body  must  have 
zero  velocity  at  that  instant.  The  body  is  then  said  to  be 
instantaneously  at  rest. 

It  may  also  be  regarded  as  geometrically  obvious  that  if 
any  two  points  0^,  O2  of  a  rigid  body  have  zero  velocity,  all 
points  of  the  line  I  joining  Oi  and  Oj  must  have  zero  velocity 
and  hence  (unless  all  points  of  tlie  ])ody  have  zero  velocity) 

83 


84  KINEMATICS  [il6. 

the  velocity  of  every  point  P  of  the  body  is  normal  to  the 
plane  {I,  P)  and  proportional  to  the  distance  of  P  from  I 
(comp.  Art.  31).  The  instantaneous  state  of  motion  of  the 
body  is  then  called  a  rotation;  the  line  I  is  called  the  instan- 
taneous axis  of  rotation;  and  the  common  factor  of  propor- 
tionality CO  of  the  velocities  is  called  the  angular  velocity. 

It  is  convenient  to  think  of  the  rotation  as  represented 
geometrically  by  a  vector  of  length  w,  laid  off  on  the  axis 
of  rotation  /,  in  a  sense  such  that  the  rotation  appears  counter- 
clockwise as  seen  from  the  arrowhead  of  the  vector  (Fig.  5, 
Art.  31).  Such  a  vector  confined  to  a  definite  straight  line  is 
called  a  localized  vector,  or  rotor.  The  rotor  co  fully  char- 
acterizes the  instantaneous  state  of  motion  of  the  body  since 
the  velocity  of  every  point  of  the  body  can  be  found  from  it 
as  we  shall  see  in  Art.  118. 

116.  The  instantaneous  state  of  motion  of  a  rigid  body  one 
of  whose  points  is  fixed,  if  not  a  state  of  rest,  is  a  rotation. 
For,  it  can  be  shown  that  if  one  point  0  of  the  body  has 
zero  velocity  there  exists  a  line  I  through  0  all  of  whose  points 
have  zero  velocity.  An  analytical  proof  is  given  in  Art.  128. 
Geometrically  the  proposition  can  be  proved  as  follows. 

Observe  first  that  in  any  motion  of  a  rigid  line  the  velocities 
of  all  points  of  the  line  must  have  equal  projections  on  the  line; 
this  follows  directly  from  the  rigidity  of  the  line.  Hence 
if  the  velocity  of  any  point  of  the  line  is  normal  to  the  line 
or  zero  the  velocities  of  all  points  of  the  line  must  be  either 
normal  to  the  line  or  zero. 

Now  consider  a  rigid  body  of  which  one  point  0  has  zero 
velocity,  and  let  Pi,  P^  be  any  tw^o  points  of  the  body,  not 
in  line  with  0.  The  velocities  of  Pi,  P2  must  be  normal  to 
OPi,  OP2,  respectively.  If  the  velocity  of  either  of  these 
points  were  zero,  the  line  joining  this  point  to  0  would  be 


117.]  VELOCITIES   IN   THE   RIGID   BODY  85 

the  required  axis  of  rotation.  We  assume  therefore  that 
these  velocities  are  both  different  from  zero.  We  can  also 
assume  that  these  velocities  are  not  parallel;  for  if  they 
happened  to  be  so  we  could  replace  one  of  the  two  points 
by  a  point  whose  velocity  is  not  parallel  to  those  of  Pj  and  P^] 
otherwise  the  motion  would  be  a  translation  which  is  im- 
possible for  a  body  with  a  fixed  point. 

It  follows  that  the  planes  through  Pi,  Po,  normal  respec- 
tively to  the  velocities  of  Pi,  P2,  must  intersect  in  a  line  I 
which  of  course  passes  through  0;  this  line  I  is  the  axis  of 
rotation.  For,  any  point  P  of  I  must  have  a  velocity  normal 
to  PO,  and  at  the  same  time  normal  to  both  PPi  and  PP2; 
this  means  that  the  velocity  of  P  is  zero. 

117.  Composition  of  intersecting  rotors.  A  rigid  body  C 
may  have,  at  a  given  instant,  an  angular  velocity  w,  about 
an  axis  li,  while  the  body  of  reference  B  to  which  li  belongs 
rotates  at  the  same  instant  with  angular  velocity  C02  about 
an  axis  h  belonging  to  a  fixed  body  A.  We  then  say  that, 
with  respect  to  A,  the  body  C  has  the  simultaneous  angular 
velocities  wi  about  U  and  coo  about  h. 

If  the  axes  U,  k  intersect,  say  at  0,  the  instantaneous  motion 
of  C  with  respect  to  A  is  a  rotation  about  an  axis  I  passing 
through  0  such  that 

sinlj.  _  smlh  _  sinZiZ2 

002  COi  CO 

with  an  angular  velocity 

CO    =     VcOi^   +   C02^  -|-   2c0lC02  COSZ1Z2. 

This  proposition,  known  as  the  parallelogram  of  angular 
velocities,  means  simply  that  two  simultaneous  angular 
velocities  coi,  coo,  al)out  intersecting  axes  h,  k,  are  together 


y 

r- 

/ 

/ 

.0^ 

/ 

_  / 

86  KINEMATICS  [118. 

equivalent  to  a  single  angular  velocity  co  about  I,  whose  rotor 
CO  is  the  geometric  sum  of  the  rotors  coi,  a;2  (Fig.  27).  The 
proof  is  as  follows. 

The  linear  velocity  of  any  point  P  of  the  body  has  two 
components,  coiri  and  co2r2,  where  ri,  r2  are  the  perpendiculars 

let  fall  from  P  on  the  axes  h, 
lo.  These  components  lie  in 
the  same  line  only  for  the 
points  of  the  plane  (^i,  k) ;  and 
they  are  equal  and  opposite 
only  for  the  points  on  the  di- 
agonal of  the  parallelogram 
constructed  on  co],  co2  as  sides. 
All  the  points  of  this  diagonal 
"■     ■  having  the  velocity  zero,  this 

line  is  the  axis  of  rotation.  The  above  equations  follow 
at  once  from  the  parallelogram  construction. 

The  proposition  is  readily  extended  to  the  composition  of 
three  or  more  angular  velocities  about  axes  passing  through 
the  same  point.  Conversely,  the  proposition  is  used  to 
resolve  a  rotor  oj  along  lines  through  any  point  of  its  line  I. 
The  resolution  of  co  along  three  rectangular  lines  through  such 
a  point  into  the  components  co^,  coy,  Wz,  so  that  co"^  =  cox^  + 
<^y'^  +  ^z^,  is  used  very  often. 

118.  In  the  case  of  rotation,  of  angular  velocity  co,  about  an 
axis  I,  if  the  motion  be  referred  to  rectangular  axes,  with  the 
origin  0  on  I,  we  can  find  the  components  Vx,  Vy,  Vz  of  the  velocity 
V  of  any  point  P{x,  y,  z)  of  the  body,  by  replacing  oo  by  its  com- 
ponents co^,  coy,  Wz  (Art.  117).  It  then  follows  from  Art.  48, 
Ex.  1,  that  Ux  produces  at  P  a  velocity  whose  components 
are  0,  —  WxZ,   coxy,  similarly,  coy  gives  the  components  UyZ, 


119. 


VELOCITIES   IN   THE   RIGID   BODY 


87 


0,  —  coyo;;  and  co,  gives  —  ui,y,w,x,  0.    Adding  the  components 
having  the  same  direction,  we  find 


WijZ  —  co^y,     Vy  =  oiz^  —  <^xZ,  v. 


<^xy 


(jOyX. 


119.  By  Art.  115,  the  velocity  y  of  P  (Fig.  28)  is  normal  to 
the  plane  (I,  P)  and  equal  to  wCP,  where  C  is  the  foot  of 
the  perpendicular  let  fall  from  P  on 
I.  Putting  OP  -  r,  4  COP  =  (/>,  so 
that  CP  =  r  sin</),  we  find 

V  =  oor  sin^. 

This  is  numerically  equal  to  the  area 
of  the  parallelogram  constructed  on 
the  vectors  co  and  r. 

In  vector  analysis  the  area  of  the 
parallelogram  of  any  two  vectors  a, 
b  is  represented  by  a  vector  c,  of 
magnitude  c  ^  ah  sin</)  (<^  being  the 
angle  between  a  and  h),  drawn  at 
right  angles  to  both  a  and  h,  in  such 
a  sense  that  a,  h,  c  form  a  right-handed  set.  This  vector 
c  is  called  the  cross-product  (vector  product,  external  pro- 
duct) of  a   and  b  and  is  denoted  by  aXb  (read  a  cross  b). 

It  then  appears  that  the  linear  velocity  v  of  P  in  our  case 
(Art.  118)  is  the  cross-product  of  the  angular  velocity  co  into 
the  radius  vector  r  of  P: 

V  =  oj  X  r. 

If  the  components  of  a,  h,  c  with  respect  to  rectangular 
axes  are  denoted  l)y  subscripts  x,  y,  z,  it  is  shown  in  vector 
analysis  that 


Fig.  28. 


tty&z     —     aj)y,  Cy     =     O^b  x     —     Oj):,  C;     =     axb  y     —     O -JO  x- 


88  KINEMATICS  [120. 

This  means  that  the  vector  equation  v  =  co  X  r  is  equivalent 
to  the  last  three  equations  of  Art.  118. 

120.  The  most  general  instantaneous  state  of  motion  of  a 
rigid  body  consists  of  a  simidtaneous  rotation  and  translation. 
For,  whatever  the  state  of  motion,  if  we  impose  on  the  whole 
body  a  velocity  of  translation  —  u  equal  and  opposite  to  the 
linear  velocity  u  of  any  one  of  its  points  0,  so  as  to  reduce 
the  velocity  of  0  to  zero,  we  have  a  body  in  a  state  of  rotation 
(Art.  116).  Hence  the  state  of  motion  of  a  rigid  body  can 
always  be  regarded  as  consisting  of  a  velocity  of  translation 
u  equal  to  the  velocity  of  any  one  of  its  points  0,  together 
with  an  angular  velocity  co  about  an  axis  I  through  0. 

121.  The  composition  of  parallel  rotors  can  be  regarded 
as  a  limiting  case  of  that  of  intersecting  rotors  (Art.  117); 

but  it  is  best  to  prove 
the  corresponding  form- 
ulae directly.  Angular  ve- 
locities about  parallel 
axes  occur,  in  particular, 
in  the  case  of  jplane  mo- 
tion of  a  rigid  body  (see  Art.  132). 

Consider  a  body  turning  with  angular  velocity  oji  about 
an  axis  Zi  (passing  through  the  point  Li,  Fig.  29,  at  right 
angles  to  the  plane  of  this  figure)  and  at  the  same  time  with 
angular  velocity  co2  about  an  axis  h  (through  L2)  parallel  to  h. 
Any  point  P  of  the  body  receives  from  wi  a  linear  velocity 
coiri  perpendicular  to  LiP  and  from  C02  a  linear  velocity  co2r2 
perpendicular  to  L^P;  the  resultant  of  these  two  is  the  total 
velocity  of  P.  The  two  components  wiri  and  wor2  fall  into 
the  same  straight  line  only  for  points  in  the  plane  {hU),  and 
their  resultant  will  be  zero  only  for  those  points  of  this  plane 
which  divide  the  distance  between  Zi  and  k  in  the  inverse 


Fig.  29. 


122.]  VELOCITIES   IN   THE   RIGID   BODY  89 

ratio  of  oji  and  co9.  In  other  words,  the  points  of  zero  velocity 
lie  on  a  straight  line  I,  parallel  to  h  and  k,  in  the  plane  (hh), 
so  situated  that  if  L  be  its  intersection  with  L1L2,  we  have 

0)1- LiL  =  0^2' LLo. 

.To  find  the  angular  velocity  w  of  the  rotation  about  I 
consider  a  particular  point,  for  instance  Lo ;  its  linear  velocity 
being  due  entirely  to  wi  about  h  is  =  ui-LiL^,  but  it  can 
also  be  regarded  as  clue  to  w  about  /;  hence 

OJy  L1L2   =   U-LLo- 

These  two  relations  give 

1j\Li  LtLi2  Lj\1j2 

CO2  COi  CO        ' 

and  as  LiL  +  LL2  =  L1L2,  we  also  have 

CO    =    COi   +   CO2. 

Thus,  the  resultant  of  two  angular  velocities  coi,  C02  about 
'parallel  axes  h,  U  is  an  angular  velocity  00  equal  to  their  algebraic 
sum,  CO  =  coi  +  C02,  about  a  parallel  axis  I  that  divides  the 
distance  between  U^Uin  the  inverse  ratio  o/co]  and  C02.  The  only 
exceptional  case,  viz.  when  coi  +  C02  =  0,  is  discussed  in  Art. 
122. 

Conversely,  an  angular  velocity  co  about  an  axis  I  can  always 
be  replaced  by  two  angular  velocities  coi,  C02  whose  sum  is  equal 
to  oj  and  whose  axes  Zi,  I2  are  parallel  to  I  and  so  selected  that  I 
divides  the  distance  between  li,  k  inversely  as  coi  is  to  C02. 

122.  The  resulting  axis  lies  between  Li  and  Lo  when  the 
components  coi,  coo  have  the  same  sense;  when  they  are  of 
opposite  sense,  it  lies  without,  on  the  side  of  the  greater  one 
of  these  components. 

If  coi  and  C02  are  equal  and  opposite,  say  coi  =  co,  coj  =  -co, 
the  resulting  axis  would  lie  at  infinity.     Two  such  equal  and 


90  KINEMATICS  1 123. 

opposite  angular  velocities  about  parallel  axes  are  said  to 
form  a  rotor-couple;  its  effect  on  the  rigid  body  is  that  of  a 
velocity  of  translation  v  =  LiL^-co  =  p-u  at  right  angles  to 
the  plane  of  the  axes.  The  distance  of  the  rotors,  L1L2 
=  p,  is  called  the  arm  of  the  couple,  and  the  product  pco  =  v 
its  moment. 

A  velocity  of  translation  v  can  therefore  always  be  re- 
placed by  a  rotor-couple  of  moment  pw  =  v,  whose  axes 
have  the  distance  p  and  lie  in  a  plane  at  right  angles  to  v. 

Again,  an  angular  velocity  co  about  an  axis  I  can  be  re- 
placed by  an  equal  angular  velocity  co  about  a  parallel 
axis  V  at  the  distance  p  from  I,  in  combination  with  a  velocity 
of  translation  v  ^  oop  at  right  angles  to  the  plane  deter- 
mined by  I  and  V. 

It  easily  follows  from  these  propositions  that  the  resul- 
tant of  any  number  of  velocities  of  translation  v,  v',  •  •  • ,  yjor- 
allel  to  the  same  plane,  and  any  number  of  angular  velocities 
CO,  cjo',  •  •  •  about  axes  perpendicular  to  this  plane  is  always  a 
single  angular  velocity  about  an  axis  perpendicular  to  the  plane 
or  a  single  velocity  of  translation  parallel  to  the  plane. 

123.  We  are  now  prepared  to  represent  in  the  most 
simple  form  the  most  general  state  of  motion  of  a  rigid  body. 
We  saw  in  Art.  120  that  it  can  be  represented  by  the  linear 
velocity  u  of  any  point  0  of  the  body,  together  vvith  an 
angular  velocity  co  about  an  axis  I  through  0. 

Let  us  resolve  ti  into  Wo  along  I  and  Ui  at  right  angles  to 
I  (Fig.  30).  In  the  plane  through  I,  perpendicular  to  Ui,  we 
can  always  (if  Wi  4=  0)  find  a  line  ?o  parallel  to  I  at  a  distance  p 
from  I  such  that  2^co  =  —  Ui;  this  line  U  is  called  the  central 
axis. 

If  we  apply  to  the  body  equal  and  opposite  angular  veloci- 
ties CO,  —  CO  about  U,  the  body  can  be  regarded  as  having 


124.] 


VELOCITIES  IN  THE  RIGID  BODY 


91 


I 


the  angular  velocity  co  about  ^o  and  the  linear  velocity  Wo 
along  U;  for,  the  rotor  couple  formed  by  co  about  I  and  —  co 
about  la  is,  by  Art.  122,  equivalent  to  a  velocity  of  trans- 
lation pco  equal  and  opposite  to  Wi 
(comp.  Art.  225). 

The  combination  of  an  angular  ve- 
locity with  a  linear  velocity  along  the 
axis  of  rotation  is  called  a  twist,  or  in- 
stantaneous screw  motion.  Thus,  the 
state  of  motion  of  a  rigid  body  at  any 
instant  is  a  twist  about  the  central  axis; 
it  may,  in  particular,  reduce  to  a  mere  '^ 
rotation,  or  to  a  translation,  or  to  a 
state  of  instantaneous  rest. 

If,  as  in  Art.  120,  we  select  an  ar- 
bitrary point  0  of  the  body  as  origin 
of  reduction,  we  obtain  a  rotor  co 
through  0  and  a  vector  u  inclined  to 
CO  at  a  certain  angle.  The  rotors  for  different  points  0  are 
always  of  the  same  magnitude,  direction,  and  sense;  but  the 
vectors  u  differ  in  general  from  point  to  point,  or  rather  from 
axis  to  axis.  If  the  origin  is  taken  on  the  central  axis  Zo, 
u  is  parallel  to  U  and  has  its  least  value,  viz.  Uq,  the  projec- 
tion of  u  on  Iq. 


pu 


Fig.  30. 


2.  Analytical  discussion. 

124.  In  studying  the  motion  of  a  rigid  body  analytically 
it  is  convenient  to  use  two  rectangular  co-ordinate  systems 
(Fig.  31),  one  Oxijz  fixed  in  space,  the  other  OiXiyiZi  fixed 
in  the  body  and  moving  with  it.  The  co-ordinates  Xi,  yi, 
Z\  of  any  point  P  of  the  body  with  respect  to  the  moving 
trihedral  are  then  constant  with  respect  to  time,  while  the 


92 


KINEMATICS 


[125. 


co-ordinates  x,  w,  z  of  the  same  point  P  with  respect  to  the 
fixed  trihedral  are  functions  of  the  time.  It  is  assumed 
throughout  that  these  functions  possess  first  and  second 
derivatives  with  respect  to  t. 

The  position  of  the  moving  trihedral  at  any  instant  is 
given  by  the  co-ordinates  a:o,  2/o,  2o  of  the  origin  Oi  and  by 


z 

Vi 

F 

/I     ,-' 

3 

Vi, 

0 

/ 

y 

/.- 

yj_ 

'y^''^^ 

• 

Fig.  31. 

the  nine  direction  cosines  of  the  axes  0\X\^  Oii/i,  OiZi  with 
respect  to  the  fixed  trihedral: 


Xi 

yi 

Zl 

X 

ai 

tti 

rts 

y 

&i 

h 

fes 

z 

Ci 

Co 

Cs 

These  12  quantities  are  functions  of  the  time. 

125.  The   ordinary  formulae   for    the    transformation   of 
rectangular  co-ordinates  give 

X  =  xo  -}-  aiXi  -\-  cioiji  +  os^i, 

y  =  yo  +  ^i-'Ci  +  boiji  +  63^1,  (1) 

2  =  20  4-  ciXi  +  coiji  +  C3Z1. 


126.]  VELOCITIES   IN   THE   RIGID   BODY  93 

It  is  well  known  that  the  9  direction  cosines  are  connected 
by  6  independent  relations  which  can  be  written  in  either 
one  of  the  equivalent  forms 

Oi^  +  bi^  +  Ci^  =  1,     0203  +  6263  +  C2C3  =  0, 

02'  +  hoj  +  02^  =  L     0301  +  6361  +  C3C1  =  0,         (2) 

03"  +  &3"  +  C3-  =  1,     aia2  +  6162  +  C1C2  =  0. 
or 

«r  +  ^2"  +  03-  =  1,    61C1  +  62C2  +  63^3  =  0, 

&r  +  62'  +  632  =  1,     cioi  +  c^ao  +  C303  =  0,        (2') 

Ci^  +  C2-  -\-  C3~  =  1,     0161  +  O262  +  0363  =  0 

The  meaning  of  these  equations  readily  appears  from  the 
meaning  of  the  angles  involved.  Thus,  the  first  equation 
expresses  the  fact  that  Oi,  61,  Ci  are  the  direction  cosines  of  a 
line,  viz.  the  axis  OiXi;  the  last  equation  expresses  the  per- 
pendicularity of  the  axes  Ox,  Oy;  and  similarly  for  the  others. 

In  mechanics,  the  two  trihedrals  are  generally  taken  as 
both  right-handed  (or  both  left-handed)  so  that  they  can 
be  brought  to  coincidence.  It  is  known  that  then  the  deter- 
minant of  the  direction  cosines  is  =  -]-  1  (and  not  —  1) 

126.  Differentiating  the  fundamental  equations  (1)  with 
respect  to  the  time  t,  we  find  for  the  compofients  of  the  velocity 
of  any  point  P  of  the  rigid  body  alo7ig  the  fixed  axes: 

X  =  xo  -\-  (hX)  +  (hyi  +  (hZi, 

y  =  yo  +  b^Xi  +  b-.yi  +  bsZi,  (3) 

i  =  io  +  CiXi  -f  c->yi  -f-  6-32:1. 

Notice  m  particular  that  if  the  motion  of  the  body  is  a 
translation,  the  direction  cosines  of  the  moving  axes  arc  con- 
stant so  that  di,  •  •  •  C3  arc  zero;  all  points  of  the  body  have 


94  KINEMATICS  [127. 

then  the  same  velocity  (i;o,  yo,  zq).  Again  if  the  point  Oi 
of  the  body  is  fixed,  Xq,  yo,  Zq  are  zero,  and  the  velocity  com- 
ponents are  linear  liomogeneous  functions  of  Xi,  yi,  Zi. 

127.  The  velocity  of  any  point  P  of  the  bod}-  relative  to 
the  ijoint  Oi  has  along  the  fixed  axes  the  components: 

X  —  ±0  =  diXi  +  d2?/i  +  ('s^i, 
y  -  yo  =  fei.Ti  +  JMji  +  6321, 
i  —  io  =  ciXi  +  c-iiji  +  i'zZi. 

To  find  the  components  along  the  moving  axes  of  this  same 
relative  velocity  of  P  we  have  only  to  project  the  components 
along  the  fixed  axes  on  the  moving  axes,  which  is  readily  done 
by  means  of  the  scheme  of  direction  cosines  in  Art.  124.  The 
resulting  expressions 

(aidi  +  6161  +  CiCi)xi  +  {aid2  +  biL  +  CiCo)?/! 

+  (aids  +  61&3  +  CiCsjZi, 

(Oidi  +  62^1  +  C2Ci)xi  +  (0002  +  62^2  +  c-2C2)yi 

+    (fl2d3    +    &2^3    +    CoCs)^!, 

(Osdl    +    fesi*!    +    C3Cl)Xl   +    (0302    +    hh    +    CsCo)//! 

+    («3d3    +    bshs    +    CsCs)^! 

can  be  simplified  very  much  l)y  means  of  the  identities  (2) 
which  give  upon  differentiation  with  respect  to  t: 

difli  +  bihi  +  CiCi  =  0, 

0202  +   ^2^2  +   <^2C2    =   0, 

d3a3  +  hsbs  +  C3C3  =  0,  .^^ 

d2a3  +  62^3  +  C2C3  =  —  ,(o2d3  +  hobs  +  C2C3), 

dsOi  +  ^3^1  +  (Vi  =  —  («3di  +  bzbi  +  C3C1), 

dia2  +  ^1^2  +  C1C2  =  —  (aid2  +  6162  +  C1C2). 

Denoting,  for  the  sake  of  brevity,  the  left-hand  members  of 

the  last  three  equations  by  coi,  un,  0^3  (we  shall  find  very  soon 

that  these  are  precisely  the  components  along  the  moving 


129.]  VELOCITIES   IN   THE   RIGID   BODY  95 

axes  of  the  rotor  w)  we  find  for  the  cotnjjonents  along  the 
moving  axes  of  the  velocity  of  P  relative  to  Oi,  the  simple  ex- 
pressions 

iC'zZi  —  wziji,     cosXi  —  wiZi,     coiiji  —  0022:1, 

which  agree  (considering  our  present  notation)  with  the 
values  found  in  Art.  118. 

128.  The  locus  of  those  points  of  the  body  whose  velocity 
relative  to  Oi  is  zero  is  given  by 

cooSi  —  CO32/1  =  0,     cosXi  —  oji^i  =  0,     a)i2/i  —  co2a;i  =  0, 

i.  e.  by 

Xi  ^  yi  ^  Zi 

COi  CO2  0)3 

This  is  a  straight  line  I  through  Oi  whose  direction  cosines  are 
proportional  to  coi,  W2,  003.  Hence  the  motion  of  the  body 
relative  to  Oi  is  a  rotation  about  the  line  I. 

To  see  that  ui,  0)2,  C03  are  the  angular  velocities  about  the 
axes  OiXi,  Oiyi,  OiZ\,  respectively,  take  Oi  as  origin  and  the 
line  I  as  axis  OiZi]  then  the  velocity  of  any  point  in  the  Xiyi- 
plane  has  the  components  —  (^^yi,  oo^Xi,  0;  i.  e.  (Art.  48, 
Ex.  1)  W3  is  the  angular  velocity  about  OiZi;  similarly  for  coi, 
Wo.  Cornp.  Art.  118.  By  Art.  117,  the  three  angular  velocities 
coi,  aj2,  W3  about  OiXi,  Oiyi,  OiZi  are  together  equivalent  to  the 
single  angular  velocity  a?  =  Vcoi^  +  C02-  +  cos^  about  the  line 
through  Oi  whose  direction  cosines  are  proportional  to  coi, 
C02,  C03. 

129.  If,  as  in  Art.  120,  we  denote  by  u  the  velocity  of  the 
point  Oi  and  l)y  Wi,  Vo,  V3  its  components  along  the  moving 
axes,  we  have  for  the  components  Vi,  V2,  Vz  of  the  absolute 
velocity  of  any  point  P  (xi,  7/1,  Z})  of  the  body  along  the  moving 
axes: 


96  KINEMATICS  1 129. 

Vi  =  Ui  +  U2Z1  —  a)3?/i, 
Vo  =  U2  -jr  0)3^:1  —  wi2i^  .       (5) 

vz  =  lis  +  carji  —  0:2X1; 
or  in  vector  notation 

V  =  u  -{-  CO  X  r. 

On  the  other  hand,  referring  the  motion  to  the  fixed  axes 
Ox,  Oy,  Oz,  let  Xo,  jjo,  Zo  be  (as  in  Arts.  125,  126)  the  co- 
ordinates with  respect  to  these  axes  of  any  point  0]  of  the 
body,  and  let  coi,  o)y,  Uz  be  the  components  of  oj  along  the 
fixed  axes;  then  the  components  of  the  absolute  velocity  of  any 
point  P{x,  y,  z)  of  the  body  along  the  fixed  axes  are 

X  =  Xo  +  coy{z  -  Zo)  -  o},{y  -  yo), 
y  =  yo  -\-  o)z{x  —  Xo)  —  o)x{z  —  Zo), 

i  =  io  +  ojx(.y  —  t/o)  —  o:y{x  —  Xo). 
If  in  these  formuljfi  we  put  a;  =  0,  ?/  =  0,  2;  =  0  we  obtain 
the  components  Ux,  Uy,  Uz  (along  the  fixed  axes)  of  the  velocity 
of  the  origin  0,  regarded  as  a  point  of  the  moving  body,  viz. 

Ux    =    Xo    —    C0y2o  +   Oizyo, 

Uy  =  yo  —  0)zXo  +  WxZo, 

Uz  ^  Zo  —  coxyo  +  Wy.ro. 
B}^  introducing  these  components  in  the  preceding  formula? 
we  obtain  for  the  components  of  the  velocity  v  of  any  point  P 
{x,  y,  z)  of  the  body  along  the  fixed  axes  the  simple  expressions 

Vx  =  X  =  Ux  -\-  ooyZ  —  cjzy, 

Vy    —    y     =    Uy    +    OJzX    —     COxZ,  (6) 

Vz.     =       Z      =      Uz     +     0>xy      —      OJyX. 

Thus  the  velocity  v  is  the  resultant  of  the  velocity  u  of  0 
{i.  e.  of  that  point  of  the  rigid  body  which  at  the  instant 
considered  happens  to  coincide  wnth  the  fixed  origin  0)  and 
of  the  linear  velocity  arising  from  the  rotation  of  angular 


131.]  VELOCITIES  IN  THE  RIGID  BODY  97 

velocity  co  about  the  line  through  0  parallel  to  the  instan- 
taneous axis. 

The  equations  (5)  and  (6)  are  of  exactly  the  same  form; 
each  of  these  sets  of  equations  is  equivalent  to  the  vector 
relation 

V  =  u  -]r  uXr] 

(5)  arises  by  projecting  on  the  moving  axes,  (6)  by  projecting 
on  the  fixed  axes. 

130.  We  have  seen  (Art.  123)  that  the  instantaneous  state 
of  motion  of  a  rigid  bod}'  is  in  general  a  twist  about  the 
central  axis  (in  the  exceptional  case  of  translation  this  line 
lies  at  infinity).  In  the  course  of  the  motion  the  central 
axis  changes  its  position  both  in  space  {i.  e.  relatively  to  the 
fixed  trihedral  Oxyz)  and  in  the  body  (relatively  to  the  moving 
trihedral  0\Xiy\Z-^.  If  the  motion  is  continuous,  the  succes- 
sive positions  of  the  central  axis  in  space  will  be  the  generators 
of  a  ruled  surface  S  fixed  in  space;  and  the  successive  posi- 
tions of  the  central  axis  in  the  body,  i.  e.  the  various  lines 
of  the  body  which  in  the  course  of  time  become  central  axes, 
will  be  the  generators  of  a  ruled  surface  Si,  fixed  in  the 
body  and  moving  with  it. 

At  any  given  instant  these  surfaces  S  and  Si  have  the 
central  axis  corresponding  to  this  instant  in  common;  it  can 
be  shown  that  they  are  in  contact  along  this  common  gen- 
erator, so  that  the  motion  consists  in  a  rotation  about,  and 
a  sliding  along,  this  generator.  Two  particular  cases,  that 
of  the  body  with  a  fixed  point  and  that  of  plane  motion, 
deserve  special  mention. 

131.  Body  with  a  fixed  point.  As  the  fixed  point  0  has 
zero  velocity  the  central  axis  is  the  instantaneous  axis  at  0, 
and  the  velocity  of  translation  is  zero.     The  surfaces  S,  Si 

8 


98 


KINEMATICS 


[132. 


are  cones  with  0  as  common  vertex;  and  the  motion  can  be 
shown  to  consist  in  the  roUing  of  »Si  over  *S.,  This  motion 
will  be  studied  more  fully  in  Chapter  XVIII. 


3.  Plane  motion. 

132.  If  the  velocities  of  all  points  of  a  rigid  Vjody  remain 
parallel  to  a  fixed  plane,  the  motion  of  the  body  is  fully 
determined  by  the  motion  of  the  cross-section  made  by 
the  bodj^  in  this  plane.  This  case  might  he  regarded  as 
the  limiting  case  of  the  motion  of  a  body  with  a  fixed  point 
as  this  point  is  removed  to  infinity.  But  it  is  more  in- 
structive to  study  it  directly. 

Taking  in  the  plane  of  the  motion  a  set  of  fixed  axes 
Ox,  Oy  (Fig.  32)  and  a  set  of  moving  axes  OiXi,  Oiiji  we  have 


Fig.  32. 


if  .To,  i/o  are  the  co-ordinates  of  Oi  and  d  is  the  angle  between 

Ox  and  Oi^r. 

X  =  Xq  -\-  Xi  cos9  —  iji  sin0, 

y  =  ijo  +  xi  sin0  +  2/i  cos^. 

133.  Differentiating  with  respect  to  t  we  find  for    the 

components  along  the  fixed  axes  of  the  velocity  v  of  P{xi,  yi) 

(xi  smd  +  yi  cos^)^  =  Xo  —  w{y  —  yo), 


(7) 


Xq 


y  =  i/o  -\-  (xi  COS0  —  yi  sin0)^  =  2/o  +  w(x  —  Xo), 


(8) 


134.]  VELOCITIES  IN  THE  RIGID  BODY  99 

where  w  =  d.  The  velocity  of  Oi  has  the  components  Xq, 
i/o;  the  velocity  of  P  relative  to  Oi  has  the  components 
—  oo(y  —  yo),  ui{x  —  Xo),  i.  e.  it  can  be  regarded  as  due  to  a 
rotation  of  angular  velocity  co  about  Oi  (Art.  48,  Ex.  1). 
The  instantaneous  motion  of  the  plane  section  of  the  body 
consists  therefore  of  a  translation  of  velocity  u{xq,  yo),  equal 
to  the  velocity  of  Oi,  and  a  rotation  about  Oi  of  angular 
velocity  co  =  6. 

Now,  excluding  the  case  of  pure  translation  when 
CO  =  0,  we  can  find  in  the  plane,  at  any  instant,  a  point  C 
of  zero  velocity,  i.  e.,  such  that 

X  =  Xo  —  u){y  —  ?/o)  =  0,     7/  =  2/0  +  o){x  —  Xo)  =  0. 

This  point  C,  the  intersection  of  the  central  axis  with  the 
plane,  is  called  the  instantaneous  center;  its  co-ordinates 
X,  y  are  evidently 

X  =  Xo ,    y  =  yo-{ .  (9) 

CO  CO 

Hence,  the  instantaneous  state  of  motion,  in  the  case  of  'plane 
motion,  is  either  a  -pure  translation  or  a  pure  rotation  about 
the  instantaneous  center. 

It  follows  that  (excepting  the  case  of  translation),  at 
any  instant,  the  velocity  of  every  point  P  is  normal  to 
the  radius  vector  CP  and  equal  to  co  times  CP.  Conversely, 
if  the  directions  of  motion  of  any  two  points  Pi,  P^  are  known, 
the  instantaneous  center  C  can  in  general  be  found  as  the 
intersection  of  the  perpendiculars  through  Pi,  P^  to  these 
directions. 

134.  In  ]-)lane  motion  the  ruled  surfaces  aS,  >Si  (Art.  130) 
are  cylind(Ts.  Instead  of  tlicse  cylinders  it  suffices  to 
consider  their  curves  of  intersection,  s,  Si  with  the  plane. 
The  curve  s  is  called  the  j&xed,  or  space,  centrode  (path  of 


100  KINEMATICS 


1135. 


the  center);  the  curve  Si  which,  as  will  be  proved  in  Art. 
135,  rolls  over  s  is  called  the  moving,  or  body,  centrode. 

Thus  any  plane  motion  consists  in  the  rolling  of  the  body 
centrode  Sx  over  the  space  centrode  s  (except  in  the  case  of 
translation).  It  is  fully  determined  if,  in  addition  to  any 
particular  position  of  these  centrodes,  the  angular  velocity 
0}  is  given  as  a  function  of  the  time. 

The  equation  of  the  space  centrode,  referred  to  the  fixed 
axes,  is  found  by  eliminating  t  between  the  equations  (9). 

That  of  the  body  centrode,  i.  e.  of  the  locus  of  those  points 
of  the  moving  figure  which  in  the  course  of  the  motion 
become  instantaneous  centers,  must  be  referred  to  the 
moving  axes  OiXi,  Oiyi.  Substituting  in  (7)  for  x,  y  the 
values  (9)  and  solving  for  Xi,  yi  we  find  the  co-ordinates 
Xi,  y\  of  the  instantaneous  center  with  respect  to  the  moving 
axes: 

Xi  =  —(xo  sin0  —  vo  cos9), 

1  (10) 

yi  =     (.to  COS0  +  yo  sin5)  ; 

CO 

the  elimination  of  t  gives  the  body  centrode. 

135.  To  prove  that,  as  stated  in  Art.  134,  the  body  cen- 
trode Si  rolls  over  the  space  centrode  s  it  suffices  to  show  that 
these  curves  have  at  the  instantaneous  center  C  not  only  a 
common  point  but  a  common  tangent ;  in  other  words,  that 
the  slopes  m,  m\  of  s,  si  at  C  are  equal.  These  slopes  can 
be  found  from  the  equations  (9)  and  (10).  From  (9)  we  find 
by  differentiating  with  respect  to  / : 

y         co.ro  +  i^'yo  —  <^Xq 
m  =  ~  =   '  — 


X      —  oiijo  -j-  co^^o  +  coyo 
Without  loss  of  generahty  we  may,  at  the  instant  considered, 


136.)  VELOCITIES   IN   THE  RIGID   BODY  101 

let  the  moving  axes  coincide  with  the  fixed  axes  and  take  the 
origin  at  the  instantaneous  center  so  that  xo,  yo,  Xo,  yo  are 
zero;  we  then  find : 

•To 

m  = . 

yo 

From  (10)  we  find  similarly 

co(xo  COS0  +  ijo  sin0)  +  co-(—  Xo  miO  +  ijo  cos6) 

^   ^Ul  = -co(.tocosg  +  yosme)  _ 

Xi      co(xo  sine  —  ijo  COS0)  +  co'^{xo  cos0  +  tjo  sin0)  ' 

—  6}(xq  sine  —  2/0  cos^) 

and,  taking  the  axes  as  above,  since  io,  ijo,  Q  are  zero: 

OCo 

nil  = . 

2/0 

Hence  w  =  Wi,  i.  e.  the  curves  s,  Si  have  a  common  tangent 
at  the  instantaneous  center. 

It  appears,  moreover,  that  this  tangent  is  norrnal  to  the 
acceleration  of  the  instantaneous  center.  Thus,  in  the  case 
of  a  circle  rolling  over  a  straight  line,  where  s  is  the  line,  Si 
the  circle,  the  acceleration  of  the  point  of  contact  is  normal 
to  the  lino. 

It  should  be  observed  that  the  equations  (9)  are  the 
parameter  equations  of  the  fixed  centrodc,  the  parameter 
lacing  t;  hence  the  ^-derivatives  .f,  y,  used  above  in  forming 
m,  are  not  the  components  of  the  velocity  of  the  instantaneous 
center  as  a  point  of  the  moving  figure  (these  velocities  are  zero), 
but  those  of  the  velocity,  say  w,  with  ivhich  the  instantaneous 
center  proceeds  along  the  curve  s.  Similarly  the  quantities 
Xi,  yi,  used  in  forming  nii,  are  the  components,  along  the 
moving  axes,  of  the  same  velocity  w. 

136.  This  velocity  w  with  which  the  instantaneous  center 
C  changes  its  position  along  the  centrodes  s,  Si  is  connected 


102 


KINEMATICS 


[136. 


by  a  simple  relation  with  the  angular  velocity  co  and  the 
radii  of  curvature  p,  pi  of  s,  Si  at  C,  viz. 

w        p       pi' 

To  prove  this  let  C  (Fig.  33)  be  the  position  of  the  instan- 
taneous center  at  the  time  /,  C  its  position  in  the  fixed  plane 


Fig.  33. 

and  Ci  its  position  in  the  moving  figure  at  the  time  t  -\-  At. 
Then,  denoting  by  As,  Asi  the  equal  arcs  CC\  CCi,  we  have 

as  definition  of  w : 

,.     As      ,.     Asi 
ic  =  hm—  =  lim  -— . 

A<=o  At       A/=o  At 

On  the  other  hand,  if  Ad  is  the  angle  through  which  any  line 
of  the  figure  turns  in  the  time  A^  we  have 

,.     A9      dd 
CO  =  inn  rr  =  37  • 

st=oAt       at 

The  motion  that  takes  place  in  the  interval  At  carries  the 


137.]  VELOCITIES   IN   THE   RIGID   BODY  103 

point  Ci  to  the  position  C  and  brings  the  normal  to  Si  at  Ci' 
to  coincidence  with  the  normal  to  s  at  C;  these  normals 
include  therefore  the  angle  Ad.  Hence  if  A(y?,  Atpi  are  the 
angles  that  these  normals  make  with  the  common  normal 
at  C  we  have  Ad  =  A(p  —  Acpi;  dividing  by  As  =  Asi  and 
passing  to  the  limit  we  find  for  the  left-hand  member 

,.      Ad      ,.      AdAt       0} 
lim  —  =  hm  —  —  =  -  , 

At=o  As      st=o  At  As      w 

provided  lim  As/At  =  w  is  4=  0.  In  the  right-hand  member, 
the  limits  of  AipjAs  and  A^fijAsi  are  clearly  the  curvatures 
of  s  and  Si  at  C;  hence 

CO  ^  1  _   1^ 

w       p       pi  * 

It  is  easily  seen  that  this  formula  holds  even  when  the 
centers  of  curvature  lie  on  opposite  sides  of  the  tangent, 
provided  we  take  pi  then  negative.  The  counterclockwise 
sense  of  co  is  taken  as  positive,  and  id  is  taken  positive  if  the 
normal  at  C  turns  counterclockwise  in  passing  to  its  new 
position  through  C 

137.  Exercises. 

(1)  A  plane  figure  moves  in  Us  plane  so  that  two  of  its  points  A,  B 
(Fig.  34)  move  along  two  perpendicular  straight  lines  Ox,  Oy. 

By  Art.  133,  the  instantaneous  center  C  is  found  as  the  intersection 
of  the  perpendiculars  at  A  to  Ox  and  at  B  to  Oy.  As  yli?  is  of  constant 
length  it  follows  readily  that  the  space  centrode  is  a  circle  of  radius 
AB  =  2a  about  0.  As  OC  =  AB  it  follows  that  the  body  centrode  is  a 
circle  of  diameter  OC  =  2a.  Hence  the  motion  can  also  be  brought 
about  by  the  rolling  of  a  circle  of  radius  a  within  a  circle  of  twice  this 
radius.  Taking  the  midpomt  Oi  oi  AB  as  origin  and  OiA  as  axis  OiXi 
of  the  set  of  moving  axes,  and  denoting  by  </>  the  angle  BAO,  we  have 
for  the  co-ordinates  of  any  point  P{x-i,  y\)  of  the  moving  figure: 

X  =  {a  -\-  Xi)  cos</)  +  ?/i  sm(/), 
2/  =  (o  —  xi)  sin<^  +  ?/i  cos(/). 


104 


KINEMATICS 


1137. 


Eliminating  4>  we  find  as  equation  of  the  path  of  P,  referred  to  the 
fixed  axes: 

fyix  -  (g  +  Xi)yY   ,    V  ViV  -  (a  -  Xi)x  V  ^ 


[{a  -  XiY  +  yi2]x2  -  402/1X2/  +  [(a  +  x^r-  +  2/i=]2/-  =  (xi=  +  2/i'  -  a')'- 

This  is  an  ellipse  referred  to  its  center.     Show  that  Oi  describes  a  circle, 
and  that  every  point  on  the  circle  about  AB  as  diameter  describes  a 


Fig. 34. 


straight  line  through  0.  Show  that  the  velocity  of  P  is  w  =  [o-  +  xx"^ 
+  2/1^  —  2a(x:  cos2<^  +  2/1  An24i)]h4>;  hence  find  the  velocities  of  B  and 
Oi  when  A  moves  uniformly. 

(2)  A  'point  A  of  the  figure  moves  along  a  fixed  straight  line  I  ivhile  a 
line  of  the  figure,  U,  containing  the  point  A,  alioays  passes  through  a  fixed 
point  B  (Fig.  35). 

The  fixed  point  B  may  be  regarded  as  the  limit  of  a  circle  which  the 
line  li  is  to  touch.  The  instantaneous  center  is  therefore  the  inter- 
section C  of  the  perpendiculars  erected  at  .A  to  Z  and  at  B  ioli. 

The  fixed  centrode  is  a  parabola  whose  vertex  is  B.  To  prove  this 
take  the  fixed  line 7  as  axis  Oy,  the  perpendicular  OB  to  it  drawn  through 


137.] 


VELOCITIES   IN   THE   RIGID   BODY 


105 


the  fixed  point  B  as  axis  Ox.     Then,  putting  OBA  =  4>  and  OB  =  a, 

we  have  for  C: 

X  =  a  -\-  y  tan</),     y  =  a  tan<^, 

whence  x  —  a  =  y^/a,  or,  with  B  as  origin  and  parallel  axes,  y"^  =  ax. 
The  proportion  y/x  —  a/y  also  follows  directly  from  the  similar  triangles 
BDC  and  AOB. 

The  equation  of  the  body  centrode,  for  A  as  origin,  AB  as  polar 
axis,  is  r  cos^^  =  a,  or  in  cartesian  co-ordinates  a^Cxi^  +  yi^)  =  xi*. 


Fig.  35. 

The  points  of  li  are  readily  seen  to  describe  conchoids;  hence  show 
how  to  construct  the  normal  at  any  point  of  a  conchoid. 

(3)  A  wheel  rolls  on  a  straight  track ;  find  the  direction  of  motion  of 
any  point  on  its  rim.     What  are  the  centrodes? 

(4)  Find  the  equations  of  the  centrodes  when  a  line  h  of  a  plane 
figure  always  touches  a  fixed  circle  while  a  point  A  of  li  moves  along 
a  fixed  line  /. 

(5)  Show  that,  in  Ex.  (4),  the  centrodes  are  parabolas  when  the 
fixed  circle  touches  the  fixed  line. 

(6)  Two  straight  lines  h,  U  of  a  plane  figure  constantly  pass  each 
through  a  fixed  point  d,  O2;  investigate  the  motion. 

(7)  Four  straight  rods  are  jointed  so  as  to  form  a  plane  quadri- 
lateral ABCD  with  invariable  sides  and  variable  angles.  One  side 
AB  being  fixed,  investigate  the  motion  of  the  opposite  side;  construct 
the  centrodes  graphically. 

(8)  A  right  angle  moves  so  that  one  side  always  passes  through  a 
fixed  point  A,  while  a  point  B  on  the  other  side,  at  the  distance  a  from 


106  KINEMATICS  [137. 

the  vertex,  moves  along  a  fixed  line  from  which  the  fixed  point  A  has 
the  distance  a;  determine  the  centrodes. 

(9)  If  the  quadrilateral  of  Ex.  (7)  be  a  parallelogram  show  that  any 
point  rigidly  connected  with  the  side  opposite  the  fixed  side  describes 
a  circle. 

(10)  One  point  A  of  a  plane  figure  describes  a  circle  while  another 
point  B  moves  on  a  straight  line  passing  through  the  center  0  of  the 
circle.  Find  the  centrodes  and  the  path  of  the  midpoint  of  AB.  Show 
how  to  construct  the  velocity  of  B  when  that  of  A  is  known. 

(11)  Two  points  Pi,  Pi  of  a  plane  figure  move  on  two  fixed  circles 
described  with  radii  ri,  ri  about  d,  Oi\  show  that  the  angular  velocities 
wi,  W2  of  0\P\,  O2P2  about  Oi,  O1  are  inversely  proportional  to  0\M,  O2M, 
M  being  the  point  of  intersection  of  O1O2  with  PjPa- 

(12)  Given  the  magnitudes  Vi,  V2  of  the  velocities  of  two  points  Pi, 
P2  of  a  plane  figure,  and  the  angle  {ih,  ih)  formed  by  their  directions; 
find  the  instantaneous  center  C  and  the  angular  velocity  w  about  C 


CHAPTER  V. 
ACCELERATIONS  IN  THE  RIGID  BODY. 

138.  The  components,  along  the  fixed  axes,  of  the  acceleration 
j  of  any  point  P(x,  y,  z)  of  a  rigid  body  are  found  by  dif- 
ferentiating with  respect  to  t  the  equations  (3)  of  Art.  126; 

this  gives: 

X  =  xo^  diXi  +  doyi  +  dzZi, 

y  =  ijo  i-  Si^^i  +  hyi  +  h^Zi,  (1) 

z  =  Zn  -{■  CiXi  +  C2?/i  +  C3Z1. 

But  we  obtain  expressions  that  are  more  easily  interpretable 
by  differentiating  the  equations  (6)  of  Art.  129.  The  first 
of  these  equations  gives 

X  =  ih  +  coyZ  —  w,y  +  CiyZ  —  o)zy, 

or,  replacing  y,  z  by  their  values  from  (G),  Art.  129, 

X    =   ilx  -{-  OiyUz   —   (jizUy  +  OiyOixy   —   Wy"X   —   Wz"£   +  C0zWx2 

+  wj,2  -  Cczy. 

Adding  and  subtracting  oix^x,  observing  that  cox^  -\-  coy"^  -\-  oi^ 
=  co^,  and  writing  down  the  expressions  for  i)  and  x  by  cyclic 
permutation  we  find : 

X     =     ilx    -\-    OJyllz    —    OizUy    -\-    Wxi'^xX    +    COy^/    4"    WjS) 

—  oi^x  +  (x>yZ  —  oizy, 

y    =   Uy  +  O^zUx   —   COxUz  +  0}y{oOxX  +  0}yy  +  OizZ) 

—  whj  -f  dzX  —  ojxZ, 

Z     =     Uz     -\-    UxUy    —     (Jiylix    +    Oiz{<JixX    +    CO  y?/    +    Oi  zZ) 

—  u'^Z    +  Cixy  —  ^yX. 

107 


108 


KINEMATICS 


[139. 


The  meaning  of  the  various  terms  will  best  appear  by  con- 
sidering some  particijlar  cases. 

139.  In  the  case  of  translation,  the  direction  cosines  of  the 
moving  axes  are  constant,  and  hence  (Art.  127)  cox,  coy,  coz,  w 
are  and  remain  zero.  Hence  the  equations  (2),  as  well  as 
(1),  reduce  to 

X  =  Ux,     ij  =  Uy,     z  =  iiz, 
as  is  otherwise  obvious  from  the  definition  of  translation, 


Fig.  36. 


140.  In  the  case  of  rotation  about  a  fixed  axis  (Fig.  36) 
we  can  take  this  axis  as  Oz  and  let  Oi  coincide  with  0,  OiZi 
with  Oz.     We  then  have 

cox  =  0,     cJx  =  0,     rro  =  0,     Ux  =  0,     iix  =  0, 

OJy     =0,  <^y     —     0,  ?/0     =     0,  Uy     =     0,  Uy    =     0, 

ojz  =  CO,     cjj  =  CO,     Zo  —  0,     Us  =  0,     iiz  =  0; 
hence  the  equations  (2)  reduce  to 


X  =  —  ixp-x  —  CO?/,  ij  =  —  oihj  +  <j^x,  z  =  (xP'Z 


w'-z 


0.    (3) 


141.]  ACCELERATIONS   IN   THE   RIGID   BODY  109 

The  acceleration  ;'  of  P  is  therefore  parallel  to  the  a:?/-plane 
and  can  be  regarded  as  consisting  of  two  components.  De- 
noting by  r  the  distance  Va:^  -\-  y"^  oi  P  from  the  fixed  axis, 
we  have  for  one  of  these  which  is  called  the  normal,  or 
centripetal,  acceleration  jn  the  components  —  wH,  —  w^?/,  0 
along  the  fixed  axes;  it  has  therefore  the  magnitude 

jn  =  wV 
and  the  direction  at  right  angles  to  the  fixed  axis,  toward  it. 
The  other  acceleration,  called  the  tangential  acceleration  jt, 
has  the  components  —  wy,  chx,  0;  it  is  tangent  to  the  circle 
described  by  P,  in  the  sense  in  which  w  increases,  and  of 

magnitude 

jt  =  <^r. 

These  results  agree  of  course  with  what  is  known  (Art.  56, 
Ex.  6)  about  the  acceleration  of  a  point  moving  in  a  circle. 

141.  Let  us  now  consider  the  important  case  of  a  rigid 
body  with  a  fixed  point.  Taking  the  fixed  point  as  fixed 
origin  0  and  letting  the  point  Oi  coincide  with  0  we  have 

Xo  =  0,     Ux  =  0,     Ux  =  0, 

yO    =    0,         Uy    =    0,         Uy    =    0, 

Zo  =  0,     ih  =  0,     u,  =  0, 
so  that  the  formulae  (2)  of  Art.  138  reduce  to 

X  =  o)x((^xX  +  Wyy  +  ojzz)  —  oo-x  +  6)yZ  —  oi^y, 

ij  =  oiy{(x)xX  +  Wyy  +  Uzz)  —  CO"!/  +  (JizX  —  co^z,         (4) 

Z    =     Wz{wxX  +   O^yy   +   <JizZ)    —    iiTZ    +   (Jixy   —   OiyX. 

The  total  acceleration  j  of  P  can  here  be  regarded  as  con- 
sisting of  three  partial  accelerations.  Denoting  by  r  the 
radius  vector  OP  =  Vx^  -\-  y^  -\-  z^  of  P  and  by  cp  the  angle 
between  r  and  the  rotor  oj,  we  have 

iOxX  +  coyT/  +  WjZ  =  a>r  cos^. 


no  KINEMATICS  [141. 

In  vector  analysis,  the  product  ah  cos^  of  any  two  vectors 
a,  b  into  the  cosine  of  the  angle  between  them  is  called  the 
dot-product  (scalar  or  internal  product)  of  the  vectors  a  and 
h  and  is  written  briefly  a  ■  b  (read :  a  dot  b) .  If  the  rectangular 
components  of  a  vector  are  denoted  by  subscripts  (as  in 
Art.  119)  we  have 

a-6  =  Uxbx  +  Oyby  +  0^62. 

Hence  in  our  case 

ajj-X  +  onylj  +  (jOzZ  =  oj-r. 

Thus  the  first  of  the  three  partial  accelerations,  ja,  has 
along  the  fixed  axes  the  components  oj^wr  costp,  ccycor  cos<p, 
Wzoor  cos<p;  it  is  therefore  represented  by  a  vector  of  length 
ja  =  co-r  C0S9?  whose  direction  is  that  of  co,  i.  e.  of  the  instan- 
taneous axis. 

The  second  partial  acceleration  jb  has  along  the  fixed  axes 
the  components  —  co^x,  —  uihj,  —  op-z;  it  is  therefore  repre- 
sented by  a  vector  of  length  ji  =  co-r,  along  r,  toward  0. 

The  third  partial  acceleration  jc  has  along  the  fixed  axes 
the  components  6:yZ  —  (h^y,  (h^x  —  oi^z,  (^xV  —  (J^yX.  It  is  there- 
fore, by  Art.  119,  the  cross-product  of  the  vectors  co  and  r;  i.e 
it  has  the  magnitude  jc  =  wr  sini/',  ^p  being  the  angle  beween 
0)  and  r,  and  it  is  perpendicular  to  both  co  and  r,  in  a  sense 
such  that  CO,  r,  j,.  form  a  right-handed  set. 

It  should  be  noted  that  each  of  the  three  partial  accelera- 
tions ja,  jb,  jc  is  a  vector  independent  of  the  co-ordinate 
system,  and  such  is  of  course  the  total  acceleration  j.  It 
follows  that  the  components  of  j  along  the  moving  axes,  if  those 
of  CO  are  denoted  by  coi,  coo,  C03,  will  be 

Xi    =    OOi(cOiXi    +    C02?/l    +    C03?l)    —    CoITi    +   COo^i    —    COgT/i, 

iji  =  co2(cO]a:i  +  coov/i  ^-  cos^i)  —  co-?/i  +  cosXi  —  WiZi,      (4') 

Zi     =    C03(cOia:i    +   C027/1   +   0032:1)    —    CO-Zi   +  COi?/i   —   C02.'Cl. 


143.] 


ACCELERATIONS   IN   THE   RIGID   BODY 


111 


In  vector  notation  we  have 

J  =  ja  +  jb  +  jc  =  (co-ri)co  -  wVi  +  ci  X  ri. 

142.  It  is  often  convenient  to  combine  the  first  two  partial 
accelerations  ja  and  jb  into  a  single  acceleration  /  which  is 
then  called  the  centripetal  acceleration. 
Now  the  vectors  ja,  jb  both  lie  in  the 
plane  {I,  P)  determined  by  the  instanta- 
neous axis  I  (through  0)  and  the  point 
P  (Fig.  37) :  ja  =  wV  costp  along  the  par- 
allel to  I  through  P,  jb  =  toV  along  PO; 
I  makes  with  OP  =  r  the  angle  99  and  ja  = 
coV  coS(^  =  jbCos<p;  hence  the  resultant  j' 
of  ja  and  jb  is 

j'  =  ja  tanv?  =  jb  siiiip  =  coV  sin^ 

along  the  perpendicular  PQ  =  r  sin^  =  r' 
let  fall  from  P  on  the  instantaneous  axis 
I.     Hence  finally 

This  centripetal  acceleration  always  exists 

(since  a  body  with  a  fixed   point  cannot 

have  a  motion  of  translation  for  which  w  =  0)  except  forsthc 

points  on  the  instantaneous  axis  for  which  r'  =  0. 

143.  The  remaining  partial  acceleration  jc  exists  only 
when  the  rotor  co  varies,  in  magnitude  or  in  direction  or  in 
both. 

Using  the  language  of  infinitesimals,  suppose  that  the  rotor 
0)  in  the  time-element  dt  receives  the  geometrical  increment 
do)  =  udt;  the  vector  a>  may  be  called  the  angular  accelera- 
tion of  the  body;  its  components  along  the  fixed  axes  are 
coj,  (Jiy,  w,.     The  body  has  therefore  the  infinitesimal  angular 


Fig.  37. 


112  KINEMATICS  [144. 

velocities  6)xdt,  dydt,  6i,dt  about  the  axes  Ox,  Oy,  Oz,  respec- 
tively. These  produce  at  P(x,  y,  z)  the  infinitesimal  linear 
velocities  0,  —  CjjZcU,  o^xydt;  Uyzdt,  0,  —  oiyxdt;  —  Uzydt, 
03zxdt,  0 ;  dividing  by  dt  and  collecting  the  terms  we  find  the 
accelerations 

(hyZ  —  Ci^y,    Oi^X  —  Wx2,    6ixy  —  (JiyX. 

which  are  the  components  of  jc- 

144.  Plane  motion.  Taking  the  plane  of  the  motion  as 
rcy-plane  we  have  to  put  co^  =  0,  Wy  =  0,  co^  =  co.  d^  =  0, 
Uy  =  0,  ojj  =  w,  Uz  =  0;  Jig  =  0  in  the  equations  (2)  of  Art. 
138  so  that  we  find 

X  =  Ux  —  wUy  —  orX  —  CO?/, 

i)  =  Uy  +  coUx  —  CO-?/  —  cox, 
while  2  =  0.     As  Ux  =  Xo  +  ojyo,  Uy  =  yo  —  ojXq  and  hence 
Ux  =  Xq  -\-  on/o  +  co?/o,  iiy  =  yo  —  wxo  —  wxo,  we  find  as  com- 
ponents of  the  acceleration  of  P(x,  y)  along  the  fixed  axes: 

X  =  xo  -  co^ix  -  xo)  -  6i{y  -  yo), 

(5) 
7j  =  i/o  -  co2(?/  -  ?/o)  +  (^{x  -  Xo). 

These  equations  are  also  obtained  directly  by  differentiating 
the  components  of  the  velocity  in  plane  motion,  (8),  Art.  133, 
which  express  that  the  instantaneous  state  of  motion  (unless 
a  translation,  co  =  0)  can  be  regarded  as  a  rotation  of  angular 
velocity  co  about  the  instantaneous  center  (xo  —  2/o/co,  yo 
+  io/co). 

The  equations  (5)  show  that  (excepting  the  case  of  transla- 
tion when  CO  =  0,  CO  =  0)  there  exists  at  every  instant  a  point  /, 
the  center  of  acceleration,  whose  acceleration  is  zero;  its 
co-ordinates  are 

co^.fo  —  coj/o  ,    o}-yo  +  cbi'o  .^s 


146. 


ACCELERATIONS   IN   THE   RIGID   BODY 


113 


145.  If  this  point  I  of  zero  acceleration  be  taken  as  origin 
of  the  moving  axes  0\Xi,  Oiiji  (Fig.  38),  the  components  along 


y 

> 

^r 

^ 

\ 

1 
1 

«i^ 

_o 

^ 

.^ 

1 

X 

^ 

Fig.  38. 

the  fixed  axes  of  the  acceleration  of  any  point  P(a:,  y)  are 
by  (5) 

-  by{x  -  x^  -  cJ(2/  -  ?/o),  -  co2(?/  -  ?/o)  +  ^{x  -  a;o). 

The  form  of  these  expressions  shows  that  if  we  put  IP  =  r, 
the  acceleration  of  P  can  be  resolved  into  a  component  coV  along 
PI  and  a  component  ur  at  right  angles  to  IP;  and  the  total 
acceleration  of  P  is 

j  =  r  Vo)^  +  oj2. 

Hence,  at  any  instant,  all  points  on  a  circle  about  I  as 
center  have  accelerations  of  equal  magnitude  and  are  equally 
inclined  to  their  radii  vectores  r  =  IP;  all  points  on  a 
straight  line  through  I  have  parallel  accelerations,  propor- 
tional to  their  radii  vectores  r  =  IP. 

146.  If  any  point  Oi  different  from  I  be  taken  as  origin 
of  the  moving  axes  (Fig.  39)  we  have  simply  to  superimpose 
its  acceleration  jo(xo,  yo);  and  it  appears  from  (5)  that  the 
acceleration  of  every  point  P  can  be  regarded  as  having  the 
three  components: 
9 


114 


KINEMATICS 


ll47. 


jo  =  the  acceleration  of  Oi, 

ji  =  oj-r  along  POi, 

jo  =  cjr  at  right  angles  to  OiP, 

where  r  =  OiP. 


Fig.  39. 


147.  If,  in  particular,  we  take  as  origin  of  the  moving  axes 
the  instantaneous  center  C  and  as  axis  OiXi  the  common 
tangent  of  the  centrodes  (Fig.  40),  the  acceleration  j  of  C 


Fig.  40. 

is  normal  to  this  tangent  (Art.  135),  and  as  CP  is  the  normal 
to  the  path  of  P  (Art.  133),  the  normal  and  tangential  com- 


148.]  ACCELERATIONS   IN   THE   RIGID   BODY  115 

ponents  of  the  acceleration  of  P  are: 

Jn^  oi^r  -  3-,    Jt  =  <^r+j—,  (7) 

where  r  —  CP.  Hence  the  loci  of  the  points  having  only 
tangential  and  only  normal  acceleration  are  the  circles: 

o^Kxi'  +  yr)  -  hi  =  0,    w{xi^  +  ^1^)  +  jxi  =  0.     (8) 

Finally,  it  can  be  shown  that  the  acceleration  of  the  instan- 
taneous center  C  is 

j  =  uw, 

where  w  is  the  velocity  with  which  the  instantaneous  center 
travels  along  the  centrodes  (Art.  135).  For,  just  as  in  Art. 
135,  we  find  by  differentiating  the  equation  (9)  of  Art.  133 
and  putting  io  =  0,  ?/o  =  0  that  the  components  of  w 
along  the  fixed  axes  are 

X  ^ ,    1/  =-  - . 

CO  CO 

whence 

^0  =  —  co.f,     .t'o  =  ooy. 

The  acceleration  of  C  is  therefore 

j  =  V.'Co^  +  2/0^  =  <^  Vi"  +  y^  =  om.  (9) 

148.  Exercises. 

(1)  A  wheel  of  radius  a  rolls  on  a  straight  track;  find  the  center  of 
acceleration :  (a)  when  the  velocity  v  of  the  axis  of  the  wheel  is  constant ; 
(b)  when  the  axis  is  uniformly  accelerated,  as  when  the  wheel  rolls 
down  an  inclined  plane. 

(2)  Find  the  locus  of  the  points  of  equal  tangential  acceleration. 

(3)  Show  that  the  components,  along  the  axes  Cxi,  Cyi  of  Fig.  40, 
of  the  acceleration  of  any  point  arej'i  =  —  co^Xi  —  o:y\,ji  =  —  co^^/i  +  wxi  + 
;■;  and  hence  the  co-ordinates  of  I  are  —  wj/(w<  +  6r),  co^j/Cw  +  w^). 
Verify  that  these  co-ordinates  satisfy  the  equations  (8) ;  this  shows  that 
the  center  of  acceleration  is  the  intersection  (different  from  C)  of  the 
circles   (8). 


116  KINEMATICS  [148. 

(4)  Show  that  the  resultant  of  j  and  cir  in  Fig.  40  is  an  acceleration 
wr',  perpendicular  to  r'  =  HP,  where  H,  the  center  of  angular  acceleration, 
is  the  intersection  of  the  circle  of  no  tangential  acceleration  (second  of 
the  equations  (8))  with  the  common  tangent  of  the  centrodes  at  C;  it 
lies  at  the  distance  CH  =  jjw  from  C.  It  follows  that  the  acceleration 
of  any  point  P  can  be  resolved  into  two  components,  uh  along  PC 
and  cor'  normal  to  HP  =  r'. 

(5)  The  first  of  the  circles  (8)  is  called  the  circle  of  inflections;  why? 

(6)  Show  that  the  diameter  of  the  circle  of  inflections  is  the  recip- 
rocal of  the  difference  of  the  curvatures  of  the  centrodes  at  their  point 
of  contact. 

(7)  Determine  the  locus  of  the  points  whose  acceleration  at  any 
instant  is  parallel:  (a)  to  the  common  normal,  (6)  to  the  common  tan- 
gent, of  the  centrodes. 


CHAPTER  VI. 
RELATIVE   MOTION. 

149.  In  studying  the  motion  of  a  point  P  relatively  to  a 
rigid  body  of  reference  B  which  is  itself  in  motion  we  use, 
just  as  in  Art.  124,  two  rectangular  trihedrals,  one  Oxyz 
fixed  in  space,  the  other  OiXiyiZi  fixed  in  the  body  B  and 
moving  with  it.  The  absolute  co-ordinates  x,  y,  z  oi  P  are 
connected  with  its  relative  co-ordinates  Xi,  yi,  Zi  by  the 
relations  (1),  Art.  125;  but  now  not  only  the  absolute  co- 
ordinates X,  y,  z  but  also  the  relative  co-ordinates  Xi,  yi,  Zi 
of  P  are  functions  of  the  time. 

Hence,  differentiating  the  equations  (1),  Art.  125,  we  find 
for  the  components,  along  the  fixed  axes,  of  the  absolute  velocity 
V  of  P: 

X  =  xo  -\-  diXi  +  (ky^  +  d^Zi  +  aiXi  +  aniji  +  as^i, 

y  =  yo-\-  hiXi  +  hojji  +  632:1  +  biXi  +  bojji  +  63^1,      (1) 

i  =  io  +  ciXi  +  62/1  +  C3Z1  +  CiXi  +  C2^i  +  csii. 

If  the  point  P  were  rigidly  connected  with  the  body  B 
the  last  three  terms  would  be  zero ;  hence  the  first  four  terms 
represent  the  components  along  the  fixed  axes  of  the  so-called 
body-velocity  Vb,  i.  e.  the  velocity  of  that  point  of  the  rigid 
body  with  which  the  point  P  happens  to  coincide  at  the 
instant  considered.  This  also  follows  from  the  equations 
(3)  of  Art.  126. 

As  ±1,  yi,  Zi  are  the  components  along  the  moving  axes  of 
the  relative  velocity  Vr  of  P  with  respect  to  B,  the  last  three 
terms  of  (1)  are  the  components  along  the  fixed  axes  of  this 

117 


118  KINEMATICS 


1150. 


same  velocity  Vr  (comp.  the  scheme  of  direction  cosines  in 
Art.  124). 

Thus  the  equations  (1)  are  merely  the  analytical  expression 
of  the  vector  equation 

V    =   Vb  -{-  Vr', 

i.  e.  the  absolute  velocity  v  of  a  point  P  is  the  geometric  sum, 
or  resultant,  of  the  body-velocity  Vb  and  the  relative  velocity  Vr', 
comp.  Art.  38. 

150.  Differentiating  the  equations  (1)  again  with  respect 
to  t  we  find  the  comjionents,  along  the  fixed  axes,  of  the  absolute 
acceleration  j  of  P. 

X  =  Xo-\-  ciiXi  +  doyi  +  d^Zi  +  2(oiii  +  (yji  +  Osii) 

+  aj-i  +  a.2yi  +  a^z,, 

y  =  yo-\-  bxXi  +  622/1  +  63^1  +  2(6i.ri  +  hill  +  6321)  ,^. 

+  biXx  +  b^iji  +  63^1, 

z  =  Zo  +  CiXi  +  Mji  +  C3Z1  +  2(^i.ri  +  c.iji  +  f'3ii) 

+  Cii-i  +  cMji  +  CiZi. 

The  first  four  terms  on  the  right  represent,  by  (1),  Art.  138, 
what  we  may  call  for  the  sake  of  brevity  the  body-acceleration 
jb,  i.  e.  the  acceleration  of  that  point  of  the  body  of  reference 
B  with  which  the  point  P  happens  to  coincide  at  the  instant 
considered.  The  last  three  terms  are  the  components  along 
the  fixed  axes  of  the  relative  acceleration  jr  of  P  whose  com- 
ponents along  the  moving  axes  are  Xi,  iji,  Zi,  i.  e.  of  the 
acceleration  of  P  relatively  to  the  moving  body  B. 

To  interpret  the  middle  terms,  those  with  the  factor  2, 
observe  that  by  comparing  Arts.  119  and  127  it  appears  that 
the  velocity  v  of  any  point  P  of  a  rigid  body  wath  a  fixed 
point  0,  which  is  a  vector  of  length  v  =  cor  sin^?,  perpendicular 
to  the  rotor  w  and  the  radius  vector  r  =  OP,  has  along  the 
fixed  axes  the  components 


150.]  RELATIVE   MOTION  119 

diXi  +  diiji  +  daZi,  biXi  +  biUi  +  63^1,  CiXi  +  Ciiji  +  CiZi. 
The  vector  that  we  wish  to  interpret  has  along  the  fixed 
axes  the  components 

di-2ii  +  d2-2?/i  +  d3-2ii,  6i-2ii  +  h2-2yi  +  63'2ii, 

ci-2xi  +  C2'2yi  +  C3-2ii; 

it  differs  from  the  preceding  vector  merely  in  having  Xi,  iji,  Zi 
replaced  by  2ii,  2?/i,  2ii.  It  represents  therefore  a  vector 
of  length  oi-2vr  sin(aj,  Vr),  at  right  angles  to  the  rotor  co  and 
the  relative  velocity  Vt{xi,  yi,  ii),  drawn  in  a  sense  such 
that  CO,  Vr,  and  this  vector  form  a  right-handed  set.  More 
briefly  we  may  say  (Art.  119)  that  this  acceleration  jc,  which 
is  called  variously  compound  centripetal,  complementary,  or 
acceleration  of  Coriolis,  is  twice  the  cross-product  of  the 
angular  velocity  co  of  the  body  B  and  the  relative  velocity 
Vr  oi  P: 

jc    =    2coX  Vr. 

Thus,  the  absolnte  acceleration  j  of  a  point  P  whose  motion 
is  referred  to  a  moving  body  of  reference  B,  is  the  geometric 
sum  of  three  accelerations,  the  body-acceleration  jh,  the  com- 
plementary acceleration  jc,  and  the  relative  acceleration  jr.' 

(3)  j    =  jh  +  jc   +  jr. 

This  proposition  is  known  as  the  theorem  of  Coriolis.  Appli- 
cations will  be  given  in  Chap.  XIX. 


PART  II:  STATICS. 


CHAPTER  VII. 

MASS;   DENSITY. 

151.  Physical  bodies  are  distinguished  from  geometrical 
configurations  by  the  property  of  possessing  mass;  and  the 
way  in  which  this  property  affects  their  motions  is  studied 
in  that  part  of  mechanics  which  is  called  dynamics. 

We  may  think  of  the  mass,  or  quantity  of  matter,  in  a 
physical  body  as  a  certain  indestructible  content  in  the 
portion  of  space  occupied  by  the  body.  By  the  methods  of 
weighing  explained  in  physics  we  can  compare  these  con- 
tents of  different  ])odies;  and,  taking  the  mass  content  of 
some  particular  body  as  the  standard  unit  we  can  express 
the  mass  of  every  body  by  a  single  real  number.  We  here 
confine  ourselves  to  so-called  gravitational  masses ;  the  num- 
ber that  expresses  such  a  mass  is  always  positive,  and  it 
remains  constant  in  whatever  way  the  body  may  move. 

The  student  must  be  warned  not  to  confound  mass  with 
weight.  The  weight  of  a  body,  as  we  shall  see  later,  is  the 
force  with  which  the  body  is  attracted  by  the  earth;  it  varies, 
therefore,  with  the  distance  of  the  body  from  the  earth's 
center,  and  would  vanish  completely  if  the  earth  were  sud- 
denly annihilated;  while  the  indestructibility  of  mass  is  the 
first  fundamental  principle  of  chemistry  and  physics. 

The  modern  developments  in  the  theory  of  electricity 
may,  and  probably  will,  lead  to  a  better  understanding  of 

120 


153.]  MASS;   DENSITY  121 

the  intimate  nature  of  mass  or  matter.  But  this  would 
hardly  affect  ordinary  mechanics  which  will  always  retain 
a  wide  range  of  applicability. 

152.  The  unit  of  mass  in  the  C.G.S.  system  (Art.  6)  is 

the  gram,  in  the  F.P.S.  system  the  'pound.     The  American 

pound  is  defined  (by  act  of  Congress,  1866)  as  2^.2^^t6  of  ^ 

kilogram : 

1  lb.  -  453.597  gm., 

1  gm.  =  0.002  204  6  lb. 

The  three  units  of  s-pace,  time,  and  mass  are  called  the 
fundamental  units  of  mechanics,  because  with  the  aid  of  these 
three,  the  units  of  all  other  quantities  occurring  in  mechanics 
can  be  expressed.  Thus  we  have  seen  how  the  units  of 
velocity  and  acceleration  are  based  on  those  of  space  and 
time,  and  we  shall  have  many  more  illustrations  in  what 
follows.  Any  unit  that  can  be  expressed  mathematically 
by  means  of  one  or  more  of  the  fundamental  units  is  called 
a  derived  unit. 

153.  A  continuous  mass  of  one,  two,  or  three  dimensions 
is  said  to  be  homogeneous  if  the  masses  contained  in  any  two 
equal  lengths,  areas,  or  volumes  (as  the  case  may  be)  are 
equal.  The  mass  is  then  said  to  be  distributed  uniformly. 
In  all  other  cases  the  mass  is  said  to  be  heterogeneous. 

The  whole  mass  ilf  of  a  homogeneous  body  divided  by 
the  space  V  it  fills  is  called  the  density  of  the  mass  or  body; 
denoting  density  by  p  we  have  therefore 

M 
P  =  y  , 

for  homogeneous  bodies.  It  follows  from  the  definition  of 
homogeneity  that  the  density  of  a  homogeneous  mass  can 


122  STATICS  [154. 

also  be  found  by  dividing  any  portion  Ailf  of  the  whole  mass 
M  by  the  space  AV  occupied  by  AM. 

In  a  heterogeneous  body,  the  quotient  AM/AV  is  called 
the  average,  or  'mean,  density  of  the  portion  AM.  The  limit 
of  this  average  density  as  the  space  AV  approaches  zero 
while  always  containing  a  certain  point  P  is  called  the  density 
of  the  mass  M  at  the  point  P: 

,.     AM      dM 

Ar=oAK        dV 

154.  The  unit  of  density  is  the  density  of  a  substance  such  that 
the  unit  of  volume  contains  tlie  unit  of  mass.  If  the  units  of  volume 
and  mass  are  selected  arbitrarily,  there  need  not  of  course  necessarily 
exist  any  physical  substance  having  unit  density  exactly.  Thus  in 
the  F.P.S.  system,  unit  density  is  the  density  of  an  ideal  substance 
one  pound  of  which  would  just  fill  a  cubic  foot.  As  a  cubic  foot  of 
water  has  a  mass  of  about  62  H  pounds,  or  1000  ounces,  the  density 
of  water  is  about  623^  times  the  unit  density. 

The  specific  density,  or  specific  gravity,  of  a  substance,  is  the  ratio 
of  its  density  to  that  qf  water  at  4°  C.  Let  p  be  the  density,  pi  the 
specific  density,  M  the  mass,  V  the  volume  of  a  homogeneous  mass, 
then  in  British  units 

M  =  pV  =  62.5pi7. 

In  the  C.G.S.  system,  the  unit  of  mass  has  been  so  selected  as  to 
make  the  density  of  water  equal  to  1  very  nearly;  in  other  words, 
the  unit  mass  (1  gram)  of  water,  at  the  temperature  of  4°  C,  fills 
one   cubic   centimeter. 

In  the  metric  system,  then,  there  is  no  difference  between  density 
and  specific  density  or  specific  gravity. 

155.  We  speak  in  mechanics  not  only  of  three-dimensional 
material  bodies,  or  volume  masses,  but  also  of  material 
surfaces,  or  surface  masses,  and  of  material  lines,  or  linear 
masses,  one  or  two  of  the  spatial  dimensions  being  made  to 
approach  zero  while  the  mass  content  remains  finite.  Thus, 
in  a  surface  mass,  sometimes  called  a  shell,  lamina,  or  mem- 
brane, a  finite  mass  content  is  assigned  to  every  finite  portion 


156.]  MASS;  DENSITY  123 

of  a  surface;  in  a  linear  mass,  often  designated  as  a  rod,  wire, 
or  chain,  a  finite  mass  content  is  assigned  to  every  finite  arc 
of  a  curve. 

If  d(x  is  the  area  element  of  the  surface  a,  ds  the  length 
element  of  the  curve  s,  the  surface  density  p  and  the  linear 
density  p"  are  defined  (comp.  Art.  153)  by 


P    = 


dM        ,,  _  dM 
da  '     ^     ~  ~di 


156.  Finally,  letting  all  three  dimensions  of  a  physical 
body  approach  zero,  while  the  mass  content  may  remain 
finite,  we  arrive  at  the  idea  of  the  mass-point,  or  particle,  viz. 
a  geometrical  point  to  which  a  definite  mass  is  assigned. 

As  a  finite  physical  mass  is  always  thought  of  as  occupying 
a  finite  space,  the  particle,  or  geometrical  point  endowed 
with  a  finite  mass,  is  a  pure  abstraction.  The  importance 
of  this  conception  lies  not  so  much  in  its  relation  to  the  idea 
that  physical  matter  is  ultimately  an  aggregation  of  such 
points  or  centers  possessing  mass  (molecules,  atoms),  but  in 
the  fact  that  for  certain  purposes  (viz.  as  far  as  translation 
only  is  concerned)  the  motion  of  a  physical  solid  is  fully 
determined  by  the  motion  of  a  certain  point  in  it,  called  the 
center  of  mass  or  centroid,  the  whole  mass  of  the  body  being 
regarded  as  concentrated  at  this  point. 


CHAPTER  VIII. 

MOMENTS   AND   CENTERS   OF   MASS. 

157.  The  product  of  a  mass  m,  concentrated  at  a  point  P, 
into  the  distance  of  the  point  P  from  any  given  point,  Hne, 
or  plane  is  called  the  moment  of  this  mass  with  respect  to  the 
point,  line,  or  plane. 

Thus,  denoting  by  r,  q,  p  the  distance  of  the  point  P  from 
the  point  0,  the  line  I,  and  the  plane  tt,  respectively,  we  have 
for  the  moments  of  m  with  respect  to  0,  I,  ir  the  expressions 
mr,  mq,  m/p. 

158.  Let  a  system  of  n  points,  or  particles,  Pi,  P2,  ■  ■  •  Pn 
be  given;  let  mi,  mo,  •  •  •  w„  be  their  masses,  and  pi,  p^,  •  ■  -pn 
their  distances  from  a  given  plane  tt.  Then  we  call  moment 
of  the  system  with  respect  to  the  plane  tt  the  algebraic  sum 

Wipi  +  mnp2  +  •  •  •  +  mnpn  =  2?np, 

the  distances  pi,  Pi,  •  •  ■  Pn  being  taken  with  the  same  sign  or 
opposite  signs  according  as  they  lie  on  the  same  side  or  on 
opposite  sides  of  the  plane  tt. 

It  is  always  possible  to  determine  one  and  only  one  distance 
p  such  that  Xmp  =  Mp,  where  M  =  Sm  =  nii  +  ?W2  +  •  •  • 
+  Wn  is  the  total  mass  of  the  system.  If  this  distance  p 
should  happen  to  be  equal  to  zero,  the  moment  of  the  system 
would  evidently  vanish  with  respect  to  the  plane  tt. 

159.  Let  us  now  refer  the  points  P  to  a  rectangular  set  of 

axes  and  let  x,  y,  z  be  their  co-ordinates.     Then  we  have  for 

the  moments  of  the  system  with  respect  to  the  co-ordinate 

planes  yz,  zx,  xy,  respectively: 

124 


161.]  MOMENTS  AND   CENTERS   OF   MASS  125 

miXi  +  ^12X2  +  •  •  •  +  ninXn  =  Smx  =  Mx, 
mii/i,  +  moijo  +  •  •  •  +  mnyn  =  ^my  =  My, 
niiZi  +  'MnZo  +  •  •  •  +  ninZn  =  '^mz  =  AIz. 
The  point  G  whose  co-ordinates  are 

_  _  I,mx       _  _  I,my      _  _  'Emz 

is  called  the  center  of  mass,  or  the  centroid,  of  the  system. 
The  centroid  is,  therefore,  defined  as  a  point  such  that  if 
the  whole  mass  M  of  the  system  he  concentrated  at  this  jjoint,  its 
moment  with  respect  to  any  one  of  the  co-ordinate  planes  is  equal 
to  the  moment  of  the  system. 

160.  It  is  easy  to  see  that  this  holds  not  only  for  the  co- 
ordinate planes  but  for  any  plane  whatever.     Jjct 

ax  -\-  ^y  -h  yz  -  po  =  0 

be  the  equation  of  any  plane  in  the  normal  form;  p,  pi, 
P2,  ■  •  •  Pn,  the  distances  of  the  points  G,  Pi,  Po,  ■  ■  ■  Pn  from 
this  plane.     Then  we  wish  to  prove  that  ^mp  =  Mp. 
Now 

p  =  ax  -\-  ^ij  -\-  yz  -  Po,     pi  =  axi  +  /3?/i  +  yZi  -  po,  -  -  ■  ; 

hence 

Iimj)  =  aXmx  +  jSSm?/  -|-  yZmz  —  poEm 
=  M{ax  +  j8?7  +  72  -  Po)  -  Mp. 

The  centroid  can  therefore  ]>e  defined  as  a  point  such  that  its 
moment  with  respect  to  any  plane  is  equal  to  that  of  the  whole 
system,  with  respect  to  the  same  plane. 

It  follows  that  the  moment  of  the  system  vanishes  for  any 
plane  passing  through  the  centroid. 

161.  In  the  case  of  a  continuous  mass,  whether  it  be  of 
one,  two,  or  three  dimensions,  the  same  reasoning  will  apply 


126  STATICS  [161. 

if  we  imagine  the  mass  divided  up  into  elements  dM  of  one, 
two,  or  three  infinitesimal  dimensions,  respectively.  The 
summations  indicated  above  by  S  will  then  become  integra- 
tions, so  that  the  centroid  of  a  continuous  mass  has  the 

co-ordinates 

fxdM  CydM  fzdM 

''''  JdM'   y~  JdM'  JdM'  ^^^ 

According  as  the  mass  is  of  one,  two,  or  three  dimensions, 
a  single,  double,  or  triple  integration  over  the  whole  mass  will 
in  general  be  required  for  the  determination  of  the  moments 
fxdM,  CydM,  fzdM  of  the  mass  with  respect  to  the  co- 
ordinate planes,  as  well  as  of  the  total  mass  JdM  =  M. 

Thus,  for  a  mass  distributed  along  a  line  or  a  curve  we 
have,  if  ds  be  the  line-element, 

dM  =  p"ds, 

where  p"  is  the  linear  density  (Art.  155);  for  a  mass  dis- 
tributed over  a  surface-area  we  have,  with  da  as  a  surface- 
element, 

dM  =  pda, 

where  p'  is  the  surface  (or  areal)  density;  finally,  for  a  mass 
distributed  throughout  a  volume  whose  element  is  rfr, 

dM  =  pdT, 

where  p  is  the  volume  density. 

If  the  mass  be  distributed  along  a  straight  line,  the  centroid 
lies  of  course  on  this  line,  and  one  co-ordinate  is  sufficient  to 
determine  the  position  of  the  centroid.  In  the  case  of  a 
plane  area,  the  centroid  lies  in  the  plane  and  two  co-ordinates 
determine  its  position;  we  then  speak  of  moments  with  re- 
spect to  lines,  instead  of  planes. 


164.]  MOMENTS   AND   CENTERS  OF   MASS  127 

162.  If  the  mass  be  homogeneous  (Art.  153),  i.  e.  if  the  density 
p  be  constant,  it  will  be  noticed  that  p  cancels  from  the  numerator 
and  denominator  in  the  equations  (2),  and  does  not  enter  into  the 
problem.  Instead  of  speaking  of  a  center  of  mass,  we  may  then  speak 
of  a  center  of  arc,  of  area,  of  volume.  The  term  ceniroid  is,  however, 
to  be  preferred  to  center,  the  latter  term  having  a  recognized  geometrical 
meaning  different  from  that  of  the  former. 

The  geometrical  center  of  a  curve  or  surface  is  a  point  such  that  any 
chord  through  it  is  bisected  by  the  point;  there  are  but  few  curves 
and  surfaces  possessing  a  center. 

The  centroid  (Art.  160)  is  a  point  such  that,  for  any  plane  passing 
through  it,  the  moment  of  the  system  is  equal  to  zero.  Such  a  point 
exists  for  every  mass,  volume,  area,  or  arc.  The  centroid  coincides, 
of  course,  with  the  center,  when  such  a  center  exists  and  the  distri- 
bution of  mass  is  uniform. 

163.  As  soon  as  p  is  given  either  as  a  constant  or  as  a  function 
of  the  co-ordinates,  the  problem  of  determining  the  centroid  of  a  con- 
tinuous mass  is  merely  a  problem  in  integration.  To  simplify  the 
integrations,  it  is  of  importance  to  select  the  element  in  a  convenient 
way  conformably  to  the  nature  of  the  particular  problem. 

Considerations  of  symmetry  and  other  geometrical  properties  will 
frequently  make  it  possible  to  determine  the  centroid  without  rcsorling 
to  integration.  Thus,  in  a  homogeneous  mass,  any  plane  of  symmetry, 
or  any  axis  of  symmetry,  must  contain  the  centroid,  since  for  such 
a  plane  or  line  the  sum  of  the  moments  is  evidently  zero. 

It  is  to  be  observed  that  the  whole  discussion  is  entirely  inde- 
pendent of  the  physical  nature  of  the  masses  rn  which  appear  here 
simply  as  numerical  coefficients,  or  "weights,"  attached  to  the  points. 
Some  of  the  masses  might  even  be  negative  provided  the  total  mass  is 
not  zero. 

It  will  be  shown  later  that  the  center  of  gravity,  as  well  as  the  center 
of  inertia,  of  a  body  coincides  with  its  centroid. 

164.  In  determining  the  centroid  of  a  given  system  it  will 
often  be  found  convenient  to  break  the  system  up  into  a  num- 
ber of  partial  systems  whose  centroids  are  either  known  or 
can  be  found  more  readily.  The  ■moment  of  the  whole  system 
is  obviously  equal  to  the  sum  of  the  moments  of  the  partial 
systems. 


128  STATICS  1165. 

Thus  let  the  given  mass  M  be  divided  into  k  partial  masses 
Ml,  •  •  •  Mk  so  that  M  =  Ml  +  •  •  ■  -\-  Mk]  let  G,  Gi,  -  ■  ■  Gk 
be  the  centroids  of  M,  Mi,  •  •  •  Mk  and  p,  pi,  •  •  •  pk  their 
distances  from  some  fixed  plane.     Then  we  have 

Mp  =  Mipi  +  •  •  •  +  Mkpk- 

165.  The  particular  case  of  two  partial  systems  occurs  most 
frequently.     We  then  have  with  reference  to  any  plane 

Mp  =  Mipi  +  M2P2,      M  ^  Mi-\~  M2. 

Letting  the  plane  coincide  successively  with  each  of  the 
three  co-ordinate  planes  it  will  be  seen  that  G  must  lie  on 
the  line  joining  Gi,  G2.  Now  taking  the  plane  at  right  angles 
to  G1G2  through  Gi  we  have 

M-GiG  =  il/o-GA; 
similarly  for  a  plane  through  G2 

M-GG2  =  Mi-GiG2U 
whence 

ljri(j  \J\J2  yjc^i 

1^2  ^  Wi  ^  ~W '' 

i.  e.  the  centroid  of  the  whole  system  divides  the  distance  of  the 
centroids  of  the  two  partial  systems  in  the  inverse  ratio  of 
their  masses. 

166.  Exercises. 

(1)  Show  that  the  centroid  of  two  particles  ?«i,  '"2  divides  their 
distance  in  the  inverse  ratio  of  the  masses  by  taking  moments  about 
the  centroid. 

Find  the  centroid: 

(2)  Of  three  masses  5,  7,  23  on  a  line,  the  mass  7  lying  midway 
between  5  and  23. 

(3)  Of  earth  and  moon,  the  moon's  mass  being  1/SO  of  that  of  the 
earth  and  the  distance  of  their  centers  240,000  miles, 

(4)  Of  three  equal  particles. 


166.]  MOMENTS  AND  CENTERS  OF  MASS  129 

(5)  Of  a  circular  arc,  radius  r,  angle  at  center  2a;  in  particular,  of 
a  semicircle. 

(6)  Of  the  arc  of  a  parabola,  if  =  4aa;,  from  vertex  to  end  of  latus 
rectum. 

(7)  Of  one  arch  of  the  cycloid  x  =  a(d  —  sinO),  y  =  ail  —  cos0). 

(8)  Of  half  the  cardioid  r  =  a(l  +  cose). 

(9)  Of  an  arc  of  the  common  helix  x  =  r  cos0,  y  =  r  sin5,  z  =  krd, 
from  6  =  0  to  e  =  d. 

(10)  Of  a  circular  arc  AB,  of  angle  AOB  =  a,  whose  density  varies 
as  the  length  of  the  arc  measured  from  A 

(11)  Show  that  the  centroid  of  a  triangular  area  is  the  intersection 
of  the  medians. 

(12)  From  a  square  ABCD  of  side  a  one  corner  EAF  is  cut  off  so 
that  AE  =  fa,  AF  =  la;  find  the  centroid  of  the  remaining  area. 

(13)  An  isosceles  right-angled  triangle  of  sides  a  being  cut  out  of 
the  area  of  its  circumscribed  circle,  find  the  centroid  of  the  remaining 
area. 

(14)  Find  the  centroid  of  the  surface  area  of  a  sphere  between  two 
parallel  planes,  by  observing  that  this  area  is  equal  to  the  surface  area 
of  the  circumscribed  cylinder  perpendicular  to  these  planes. 

(15)  Show  that  for  an  area  a,  bounded  by  a  curve  y  —  fix),  the 
axis  Ox  and  two  ordinates,  we  have 


X  =   I     xydx,    <T-y  =  I  \     y'^dx; 


and  hence  find  the  centroid:  (a)  of  the  area  bounded  by  the  parabola 
2/2  =  4ax,  the  axis  Ox  and  an  ordinate;  (5)  of  the  area  between  the 
curve  y  =  sina;  from  a;  =  0  to  .r  =  tt  and  the  axis  Ox ;  (c)  of  a  quadrant 
of  an  ellipse;  (d)  of  the  segment  cut  off  from  an  ellipse  by  the  chord 
joining  the  extremities  of  the  axes. 

(16)  Show  that  for  the  area  a,  bounded  by  a  curve  r  =  j{&)  and  two 
of  its  radii  vectores,  we  have 


=  l  \      r3  cosedO,     (7-  y  =  t   }      r^  f- 


smBde. 


(17)  Find  the  centroid  of  the  sector  of  a  circle,  radius  r,  angle  at 
center  2a. 

(18)  A  bowl  in  the  form  of  a  licmisph(>r(!  is  closed  l)y  a  circular  lid 
of  a  material  whose  density  is  three  tinu>s  that  of  the  bowl.  Find  the 
centroid. 

10 


130  STATICS  [166. 

(19)  The  cissoid  (2a  —  x)y^  =  x^  can  be  represented  by  the  equa- 
tions X  =  2a  sin^,  y  =  2a  sin^^/cos^,  where  d  is  the  polar  angle,  2a 
the  distance  from  cusp  to  asymptote.  Show  that  the  centroid  of  the 
area  between  the  curve  and  its  asymptote  divides  the  distance  between 
cusp  and  asymptote  in  the  ratio  5:1. 

(20)  The  centroid  of  a  rectilinear  segment  of  length  I  whose  linear 
density  is  proportional  to  the  «th  power  of  the  distance  from  one  end 
is  at  the  distance  {n  +  1)1/ (n  +  2)  from  that  end.  Hence  show  that 
(o)  the  centroid  of  a  triangular  area  lies  on  the  median  at  %  the  distance 
from  the  vertex  to  the  base;  (b)  the  centroid  of  the  surface  area  of  a 
cone  or  pyramid  lies  on  the  line  joining  the  vertex  to  the  centroid  of  the 
base,  at  %  the  distance  from  the  vertex  to  the  base;  (c)  the  centroid 
of  the  volume  of  a  cone  or  pyramid  lies  on  the  same  line,  at  %  the 
distance  from  the  vertex  to  the  base. 

(21)  For  a  solid  of  revolution,  generated  by  the  revolution  of  the  curve 
y  =  f(x)  about  the  axis  of  x  and  bounded  by  planes  perpendicular  to 
the  axis  Ox,  show  that  the  centroids  of  the  curved  surface  area  a  and 
of  the  volume  t  are  given  by: 

a-xa  =  2w  \   '  xy  1^1  +  w'^  dx,     T  ■  Xt  =  IT  \     xyHx. 

(22)  Find  the  centroid  of  the  segment  of  a  sphere  between  two 
parallel  planes;  and  hence  (a)  that  of  a  segment  of  height  A,  cut  off  by 
a  plane;  (b)  that  of  a  hemisphere;  (c)  that  of  a  spherical  sector  of  ver- 
tical angle  2a. 

(23)  Find  the  centroid  of  the  paraboloid  of  revolution  of  height  h, 
generated  by  the  revolution  of  the  parabola  y~  =  4ax  about  its  axis. 

(24)  The  area  bounded  by  the  parabola  y-  =  4ax,  the  axis  of  x,  and 
the  ordinate  y  =  yi  revolves  about  the  tangent  at  the  vertex.  Find  the 
centroid  of  the  solid  of  revolution  so  generated. 

(25)  The  same  area  as  in  Ex.  (6)  revolves  about  the  ordinate  yi. 
Find  the  centroid. 

(26)  Find  the  centroid  of  an  octant  of  an  ellipsoid 

xVa^  +  yy¥  +  zVc^  =  1. 

(27)  The  equations  of  the  common  cycloid  referred  to  a  cusp  as 
origin  and  the  base  as  axis  of  x  are  x  =  a(d  —  sinO),  y  =  a{l  —  cos^). 
Find  the  centroid:   (a)  of  the  arc  of  the  semi-cycloid  {i.  e.  from  cusp 


166.1  MOMENTS  AND   CENTERS   OF   MASS  131 

to  vertex) ;  (b)  of  the  plane  area  included  between  the  semi-cycloid  and 
the  base;  (c)  of  the  surface  generated  by  the  revolution  of  the  semi- 
cycloid  about  the  base;  (d)  of  the  volume  generated  in  the  same  case. 
(28)  Find  the  centroid  of  a  solid  hemisphere  whose  density  varies 
as  the  nth  power  of  the  distance  from  the  center. 


CHAPTER  IX. 
MOMENTUM  ;   FORCE  ;   ENERGY. 

167.  Let  us  consider  a  point  moving  with  constant  accelera- 
tion from  rest  m  a  straiglit  line.  We  know  from  Kinematics 
(Art.  16)  that  its  motion  is  determined  by  the  eauations 

V  =  jt,   s  =  ^jr~,   ^v''  -  js,  (1) 

where  s  is  the  distance  passed  over  in  the  time  t,  v  the  velocity, 
and  j  the  acceleration,  at  the  time  t. 

If,  now,  for  the  single  point  we  sul^stitute  an  m-tuple  point, 
i.  e.  if  we  endow  our  point  with  the  mass  ni,  and  thus  make  it 
a,  particle  (see  Art.  156),  the  equations  (1)  must  be  multiplied 
by  m,  and  we  obtain 

mv  =  mjt,     ms  =  h^njt},     \mv'^  =  mjs.  (2) 

The  quantities  mv,  7nj,  iww"  occurring  in  these  equations 
have  received  special  names  because  they  correspond  to 
certain  physical  conceptions  of  great  importance. 

168.  The  product  mv  of  the  mass  m  of  a  particle  into  its  veloc- 
ity V  is  called  the  momentum,  or  the  quantity  of  motion,  of  the 
particle. 

In  observing  the  behavior  of  a  physical  body  in  motion,  we  notice 
that  the  effect  it  produces — for  instance,  when  impinging'  on  another 
body,  or  more  generally,  whenever  its  velocity  is  changed — depends 
not  only  on  its  velocity,  but  also  on  its  mass.  FamiUar  examples  are 
the  following :  a  loaded  railroad  car  is  not  so  easily  stopped  as  an  empty 
one;  the  destructive  effect  of  a  cannon-ball  depends  both  on  its  velocity 
and  on  its  mass;  the  larger  a  fly-wheel,  the  more  difficult  is  it  to  give  it 
a  certain  velocity;  etc. 

It  is  from  experiences  of  this  kind  that  the  physical  idea  of  mass  is 

derived. 

132 


I7i.[  MOMENTUM;   FORCE;  ENERGY  133 

The  fact  that  any  change  of  motion  in  a  physical  body  is  affected  by 
its  mass  is  sometimes  ascribed  to  the  so-called  "inertia,"  or  "force  of 
inertia,"  of  matter,  which  means,  however,  nothing  else  but  the  property 
of  possessing  mass. 

169.  Momentum,  being  by  definition  (Art.  168)  the  product  of 
mass  and  velocity,  has  for  its  dimensions  (Art.  6), 

MV  =  MLT-\ 

The  unit  of  momentum  is  the  momentum  of  the  unit  of  mass  having 
the  unit  of  velocity.  Thus  in  the  C.G.S.  system  the  unit  of  momentum 
is  the  momentum  of  a  particle  of  1  gram  moving  with  a  velocity  of  1 
cm.  per  second.  In  the  F.P.S.  system,  the  unit  is  the  momentum  of  a 
particle  of  1  pound  mass  moving  with  a  velocity  of  1  ft.  per  second. 

To  find  the  relations  between  these  two  units,  let  there  be  x  C.G.S. 
units  in  the  F.P.S.  unit;  then  \ 

gm.  cm.       .,     lb.  ft. 
X  •  =  1 ; 

sec.  sec. 

hence 

^   Ib^     ft. 
gm. '  cm. ' 
or,  by  Art.  152  and  Art.  6, 

X  =  13,825.7; 

i.  e.  1  F.P.S.  unit  of  momentum  =  13,825.7  C.G.S.  units,  and  1  C.G.S. 
unit  =  0.000072  33  F.P.S.  units. 

170.  Exercises. 

(1)  What  is  the  momentum  of  a  cannon-ball  weighing  200  lbs.  when 
moving  with  a  velocity  of  1500  ft.  per  second? 

(2)  With  what  velocity  must  a  railroad-truck  weighing  3  tons  move 
to  have  the  same  momentum  as  the  cannon-ball  in  Ex.  (1)? 

(3)  Determine  the  momentum  of  a  one-ton  ram  after  falling  through 
4  feet. 

171.  The  'product  mj  of  the  mass  m  of  a  particle  into  its 
acceleration  j  is  called  force.  Denoting  it  by  F,  we  may 
write  our  equations  (2)  in  the  form 

mv  =  Ft,     s  =  *      t^,     hnv'^  =  Fs.  (3) 

"  m 


134  STATICS 


.173] 


As  long  as  the  velocity  of  a  particle  of  constant  mass 
remains  constant,  its  momentum  remains  unchanged.  If  the 
velocity  changes  uniformly  from  the  value  v  at  the  time  t 
to  v'  at  the  time  t',  the  corresponding  change  of  momentum  is 

my'  —  mv  =  mji'  —  mjt  =  F{t'  —  t);  (4) 

hence 

„      mv'  —  mv  ,^. 

Here  the  acceleration,  and  hence  the  force,  was  assumed 
constant.  If  F  be  variable,  we  have  in  the  limit  as  t'  —  i 
approaches  zero: 

F=^  =  m^.  (6) 

at  at 

Instead  of  defining  force  as  the  product  of  mass  and 
acceleration,  we  may  therefore  define  it  as  the  rate  of  change 
of  momenturyi  with  the  time. 

172.  Integrating  equation  (6),  we  find 

J^  Felt  =  mv'  —  mv.  (7) 

The  'product  F(t'  —  t)  of  a  constant  force  into  the  time  t'  —  t 
during  which  it  acts,  and  in  the  case  of  a  variable  force,  the 
time-integral  J  Fdt,  is  called  the  impulse  of  the  force  during 
this  time. 

It  appears  from  the  equations  (4)  and  (7)  that  the  impulse 
of  a  force  during  a  given  time  is  equal  to  the  change  of  momentum 
during  that  time. 

173.  The  idea  of  force  is  no  doubt  primarily  derived  from  the  sensa- 
tion produced  in  a  person  by  the  exertion  of  his  "muscular  force." 
Like  the  sensations  of  Hght,  sound,  heat,  etc.,  the  sensation  of  exerting 
force  is  capable,  in  a  rough  way,  of  measurement.  But  the  physiological 
and  psychological  phenomena  attending  the  exertion  of  muscular  force 
when  analyzed  more  carefully  are  very  complicated. 


174-1  MOMENTUM;   FORCE;   ENERGY  135 

In  popular  language  the  term  "force"  is  applied  in  a  great  variety 
of  meanings.  For  scientific  purposes  it  is  of  course  necessary  to  attach 
a  single  definite  meaning  to  it. 

It  is  customary  in  physics  to  speak  of  force  as  ^producing  or  generating 
velocity,  and  to  define  force  as  the  cause  of  acceleration.  Thus  obser- 
vation shows  that  the  velocity  of  a  falling  body  increases  during  the 
fall;  the  cause  of  the  observed  change  in  the  velocity,  i.  e.  of  the  ac- 
celeration, is  called  the  force  of  attraction,  and  is  supposed  to  be  exerted 
by  the  earth.  Again,  a  body  falling  in  the  air,  or  in  some  other  medium, 
is  observed  to  increase  its  velocity  less  rapidly  than  a  body  falling 
in  vacuo;  a  force  of  resistance  is  therefore  ascribed  to  the  medium  as  the 
cause  of  this  change.  In  a  similar  way  we  speak  of  the  expansive  force 
of  steam,  of  electric  and  magnetic  forces,  etc.,  because  it  is  convenient 
to  think  of  such  agencies  as  producing  changes  of  velocity. 

Now,  any  change  in  the  velocity  v  of  a  body  of  given  mass  m  implies 
a  change  in  its  momentum  mv;  and  it  is  this  change  of  momentum,  or 
rather  the  rate  at  which  the  momentum  changes  with  the  time,  which 
is  of  prime  importance  in  all  the  applications  of  mechanics.  It  is  there- 
fore convenient  to  have  a  special  name  for  this  rate  of  change  of  mo- 
mentum, and  that  is  what  is  called  force  in  mechanics. 

Thus,  in  using  this  term  "force,"  it  is  not  intended  to  assert  any- 
thing as  to  the  objective  reality  or  actual  nature  of  force  and  matter  in 
the  popular  acceptation  of  these  terms.  With  the  ultimate  causes 
science  has  nothing  to  do;  it  can  observe  only  the  phenomena  them- 
selves. 

174.  The  definition  of  force  (Art.  171)  as  the  product  of  mass  and 
acceleration  gives  the  dimensions  of  force  as 

F  =  MJ  =  MLT-^. 

The  tmit  of  force  is  therefore  the  force  of  a  particle  of  unit  mass 
moving  with  unit  acceleration. 

Hence,  in  the  C.G.S.  system,  it  is  the  force  of  a  particle  of  1  gram 
moving  with  an  acceleration  of  1  cm./sec.^.  This  unit  force  is  called 
a  dyne. 

The  definition  is  sometimes  expressed  in  a  slightly  different  form. 
We  may  say  the  dyne  is  the  force  which,  acting  on  a  gram  uniformly 
for  one  second,  would  generate  in  it  a  velocity  of  1  cm. /sec;  or  would 
give  it  the  C.G.S.  unit  of  acceleration;  or  it  is  the  force  which,  acting 


136  STATICS  [175. 

on  any  mass  uniformly  for  one  second,  would  produce  in  it  the  C.G.S. 
unit  of  momentum. 

That  these  various  statements  mean  the  same  thing  follows  from 
the  fundamental  formulae  F  =  mj,  v  =  jl,  if  F,  m,  t,  v,  j  be  expressed 
in  C.G.S.  units. 

In  the  F.P.S.  system,  the  unit  of  force  is  the  force  of  a  mass  of 
1  lb.  moving  with  an  acceleration  of  1  ft./sec.^.     It  is  called  the  poundal. 

175.  To  find  the  relation  between  these  two  units,  let  x  be  the 
number  of  dynes  in  the  poundal;  then  we  have 

gm.  cm.       ^     lb.  ft. 

X  ■ ;: —    =    1 


860.-=  sec.'' 

hence,  just  as  in  Art.  169,  x  =  13,825.7;  i.  e.  1  poundal  =  13,825.7 
dynes,  and  1  dyne  =  0.000  072  33  i)oundals. 

176.  Another  system  of  measuring  force,  the  so-called  gravitation 
(or  engineering)  system,  is  in  very  common  use,  and  must  be  explained 
here. 

Among  the  forces  of  nature  the  most  common  is  the  force  of  gravity, 
or  the  weight,  i.  e.  the  force  with  which  any  physical  body  is  attracted 
by  the  earth.  As  we  have  convenient  and  accurate  appliances  for 
comparing  the  weights  of  different  bodies  at  the  same  place,  the  idea 
suggests  itself  of  selecting  as  unit  force  the  weight  of  a  certain  standard 
mass. 

In  the  metric  gravitation  system  the  weight  of  a  kilogram  has  been 
selected  as  unit  force;  in  the  British  gravitation  system  the  weight 
of  a  pound  is  the  unit  force. 

177.  The  system  in  which  the  units  of  time,  length,  and  mass  are 
taken  as  fundamental,  while  the  unit  of  force  ( =  mass  times  accelera- 
tion) is  regarded  as  a  derived  unit  (Art.  175),  is  called  the  absolute  or 
scientific  system,  to  distinguish  it  from  the  gravitation  system  (Art. 
176)  in  which  the  units  of  time,  length,  and  force  are  taken  as  funda- 
mental, while  the  unit  of  mass  (=  force  divided  by  acceleration)  is  a 
derived  unit. 

As  the  weight  of  a  body  varies  from  place  to  place  with  the  variation 
of  the  acceleration  of  gravity  g,  the  unit  of  force  as  defined  in  Art.  176 
would  not  be  constant.  This  difficulty  can  be  avoided  by  defining  the 
unit  of  force  as  the  weight  of  a  kilogram  or  pound  at  some  definite  place, 
say  at  London,  or  in  latitude  45°  at  sea  level.     With  this  modification, 


180.]  MOMENTUM;   FORCE;   ENERGY  137 

the  gravitation  system  desenves  the  name  of  an  absolute  system  as  much 
as  does  the  system  in  which  mass  is  the  thii-d  fundamental  unit. 

The  general  equations  of  mechanics  are  of  course  independent  of 
the  system  of  measurement  adopted;  they  hold  as  well  in  the  gravita- 
tion as  in  the  scientific  or  absolute  system.  In  the  present  work  the 
language  of  the  latter  system  is  generally  used  in  the  text  (not  always 
in  the  exercises).  This  system,  since  its  introduction  by  Gauss  and 
Weber,  has  found  general  acceptance  in  scientific  .work. 

In  statics  where  we  are  mainly  concerned  with  the  ratios  of  forces 
and  not  with  their  absolute  values  it  rarely  makes  any  difference 
which  system  is  used  provided  all  forces  are  expressed  in  the  same 
unit.  And  as  elementary  statics  deals  largely  with  the  effects  of  gravity, 
the  gravitation  system  is  often  used  in  statical  problems. 

178.  The  numerical  relation  between  the  scientific  and  gravitation 
measures  of  force  is  expressed  by  the  equations 

1  kilogram  (force)  =  1000  g  dynes, 
1  pound  (force)  =  g  poundals, 

where  g  is  about  981  in  metric  units,  and  about  32.2  in  British  units. 
In  most  eases  the  more  convenient  values  980  and  32  may  be  used. 

179.  Exercises. 

(1)  What  is  the  exact  meaning  of  "a  force  of  10  tons"?  Express 
this  force  in  poundals  and  in  dynes. 

(2)  Reduce  2,000,000  dynes  to  British  gravitation  measure. 

(3)  Express  a  pressure  of  2  lbs.  per  square  inch  in  kilograms  per 
square  centimeter. 

(4)  Show  that  a  poundal  is  very  nearly  half  an  ounce,  and  a  dyne 
a  little  over  a  milligram,  in  gravitation  measure. 

180.  The  quantity  \mv'^,  i.  e.  half  the  product  of  the  mass  of 
a  particle  into  the  square  of  its  velocity,  is  called  the  kinetic 
energy  of  the  particle. 

Let  us  consider  again  a  particle  of  constant  mass  ?n  moving 
with  a  constant  acceleration,  and  hence  with  a  constant 
force;  let  v  be  the  velocity,  s  the  space  described,  at  the  time  i; 
y',  s'  the  corresponding  values  at  the  time  t'.     Then  the  last 


138  STATICS  [182. 

of  the  three  fundamental  equations  (see  Arts.  167  and  171) 

gives 

^v''~  -  hnv-  =  F{s' -  s);  (8) 

hence 

F  = -, — .  (9) 

s  —  s 

If  F  be  variable,  we  have  in  the  limit 

F  =  — ^^-^ — -  =  mv  ^  .  (10) 

as  as 

Force  can  therefore  be  defined  as  the  rate  at  ivhich  the 
kinetic  energy  changes  with  the  space.  (Compare  the  end  of 
Art.  171.) 

181.  Integrating  the  last  equation  (10),  we  find 

£'Fds  =  hnv'^  -  hnv'-.  (11) 

The  product  F{s'  —  s)  of  a  constant  force  F  into  the  space 
s'  —  s  described  in  the  direction  of  the  force,  and  in  the  case 
of  a  variable  force,  the  space-integral  f  Fds,  is  called  the 
work  of  the  force  for  this  space. 

The  equations  (8)  and  (11)  show  that  the  work  of  a  force  is 
equal  to  the  corresponding  change  of  the  kinetic  energy. 

We  have  here  assumed  that  the  force  acts  in  the  direction 
of  motion  of  the  particle.  A  more  general  definition  of  work 
including  the  above  as  a  special  case  will  be  given  later  (Art. 
261). 

The  ideas  of  energy  and  work  have  attained  the  highest 
importance  in  mechanics  and  mathematical  physics  within 
comparatively  recent  times.  Their  full  discussion  belongs 
to  Kinetics  (see  Part  III). 

182.  According  to  their  definitions,  both  momentum  (Art. 
168)  and  force  (Art.  171)  may  be  regarded  mathematically 


183.]  MOMENTUM;  FORCE;  ENERGY  139 

as  mere  numerical  multiples  of  velocity  and  acceleration, 
respectively.  They  are  therefore  so-called  vector-quantities; 
i.  e.  a  momentum  as  well  as  a  force  can  be  represented  geo- 
metrically by  a  segment  of  a  straight  line  of  definite  length, 
direction,  and  sense.  Moreover,  as  they  are  referred  to  a 
particular  point,  viz.,  to  the  point  whose  mass  is  in,  the  line 
representing  a  momentum  or  a  force  must  be  drawn  through 
this  point;  the  hne  has  therefore  not  only  direction,  but  also 
position;  i.  e.  a  momentum  as  well  as  a  force  is  represented 
geometrically  hy  a  rotor  (compare  Art.  115). 

It  follows  that  concurrent  forces,  for  instance,  can  be  com- 
pounded l^y  geometrical  addition,  as  will  be  explained  more 
fully  in  Chapter  X. 

On  the  other  hand,  kinetic  energy  and  work  are  not  vector- 
quantities. 

183.  The  ideas  of  momentum,  force,  energy,  work,  with  the  funda- 
mental equations  connecting  them,  as  given  in  the  preceding  articles, 
form,  the  groundwork  of  the  whole  science  of  theoretical  dynamics. 
The  application  of  this  science  to  the  interpretation  of  natural  phenom- 
ena gives  results  in  close  agreement  with  observation  and  experiment. 
It  is  therefore  important  to  inquire  what  are  the  physical  assumptions 
and  experimental  data  on  which  this  application  of  dynamics  is  based. 

These  assumptions  were  formulated  with  remarkable  clearness  by 
Sir  Isaac  Newton  m  his  Philosophioe  naturalis  prindpia  malhematica, 
first  published  in  1687,  and  have  since  been  known  as  Newton's  laws 
of  motion.  As  these  three  axiomata  sive  leges  motus,  as  Newton  terms 
them,  are  very  often  referred  to  and,  at  least  bj^  English  writers  on 
dynamics,  are  usually  laid  down  as  the  foundation  of  the  science,  they 
are  given  here  in  a  literal  translation: 

I.  Every  body  persists  in  its  state  of  rest  or  of  uniform  motion  along 
a  straight  line,  except  in  so  far  as  it  is  compelled  by  impressed  {i.  e. 
external)  forces  to  change  that  state. 

II.  Change  of  motion  is  proportional  to  the  impressed  moving  force 
and  takes  place  along  the  straight  line  in  which  that  force  acts. 


140  STATICS  [184 

III.  To  every  action  there  is  an  equal  and  contrary  reaction;  or, 
the  mutual  actions  of  two  bodies  on  one  another  are  always  equal  and 
directed  in  contrary  senses. 

184.  Some  explanation  is  necessary  to  understand  correctly  the 
meaning  of  these  laws.  Indeed,  Newton's  laws  should  not  be  studied 
by  themselves;  they  become  intelUgible  only  if  taken  in  connection 
with  the  definitions  preceding  them  in  the  Prindpia,  and  with  the  ex- 
planations and  corollaries  that  Newton  himself  has  appended  to  them. 

The  word  'body"  must  be  taken  to  mean  particle;  the  word  "mo- 
tion" in  the  second  law  means  what  is  now  called  momentum. 

All  three  laws  imply  the  idea  of  force  as  the  cause  of  any  change  of 
momentum  in  a  particle. 

185.  With  this  definition  of  force  the  first  law,  at  least  in  the  ordi- 
nary form  of  statement,  for  a  single  particle,  merely  states  that  where 
there  is  no  cause  there  is  no  effect.  While  this  law  may  appear  super- 
fluous to  us,  it  was  not  so  in  the  time  of  Newton.  Kepler  and  Galileo, 
less  than  a  century  before  Newton,  were  the  first  to  insist  more  or  less 
clearly  on  this  so-called  law  of  inertia,  viz.  that  there  is  no  intrinsic 
power  or  tendency  in  moving  matter  to  come  to  rest  or  to  change  its 
motion  in  any  way. 

186.  The  second  law  gives  as  the  measure  of  a  constant  force  the 
amount  of  momentum  generated  in  a  given  time  (see  Art.  171);  it 
can  be  called  the  law  of  force.  If  force  be  defined  as  the  cause  of  any 
change  of  momentum,  the  second  law  follows  naturally  by  assuming,  as 
is  usually  done,  that  the  effect  is  proportional  to  the  cause. 

The  first  two  laws  may  thus  be  regarded  from  the  mathematical 
point  of  view  as  nothing  but  a  definition  of  force;  but  they  are  certainly 
meant  to  emphasize  the  phj'sical  fact  that  the  assumed  definition  of 
force  is  not  arbitrary,  but  based  on  the  characteristics  of  motion  as 
observed  in  nature. 

In  the  corollaries  to  his  laws  Newton  tries  to  show  how  the  compo- 
sition and  resolution  of  forces  by  the  parallelogram  rule  follows  from 
his  definition.  In  deriving  this  result  he  tacitly  assumes  that  the  action 
of  any  force  on  a  particle  takes  place  independentlj'  of  the  action  of 
any  other  forces  that  may  be  acting  on  the  particle  at  the  same  time, 
a  principle  that  would  seem  to  deserve  explicit  statement.  Some 
writers  on  mechanics  prefer  to  replace  Newton's  second  law  by  this 
principle  of  the  independence  of  the  action  of  forces. 


187.]  MOMENTUM;  FORCE;  ENERGY  141 

187.  The  third  law  expresses  the  physical  fact  that  in  nature  all 
forces  occur  in  pairs  of  equal  and  opposite  forces.  Two  such  equal  and 
opposite  forces  in  the  same  line  are  often  said  to  constitute  a  stress. 
Newton's  third  law  has  therefore  been  called  the  law  of  stress. 

This  law,  which  was  first  clearly  conceived  in  Newton's  time,  involves 
what  may  be  regarded  as  the  second  fundamental  property  of  matter 
or  mass  (the  first  being  its  indestructibility) ;  viz.  that  any  two  particles 
of  matter  determine  in  each  other  oppositely  directed  accelerations  along 
the  line  joining  them. 


CHAPTER  X. 

STATICS  OF  THE   PARTICLE. 

1.88.  According  to  the  definition  of  force  (Arts.  171,  173), 
a  single  force  F  acting  on  a  particle  of  mass  m  produces  ari 
acceleration  j  such  that  F  =  mj;  i.  e.  the  vector  F  is  m 
times  the  vector  j. 

If  two  forces  Fi,  Fo  act  on  the  same  particle,  it  is  assumed 
(Art.  186)  that  each  acts  as  if  the  other  were  not  present; 

/ -z^     hence,  if  ju  jo  are  the  ac- 

/  ^^  I      celerations    which    Fi,    Fo 

/  *  ^-^^^^  /       would    produce  separately, 

•l ^"^      j  I         then  the  combined  effect  of 

J^      ^         FJ  Fi  and  Fo  will  be  to  produce 

.^.  an  acceleration  equal  to  the 

Fig.  41.  ,  ^     . 

resultant,  or  geometric  sum, 

i  =  Ji  +  J2,  of  the  accelerations  ji,  j^;  and  this  resultant  ac- 
celeration j  can  be  produced  by  a  single  force  R  =  mj  (Fig. 
41). 

The  combined  effect  of  the  two  forces  Fi,  F2  acting  on  the 
same  particle  m  is  thus  the  same  as  that  of  that  single  force 
R  which  is  the  resultant,  or  geometric  sum,  of  Fi  and  F2. 
The  two  forces  Fi,  Fo  are  said  to  be  equivalent  to  the  single 
force  R;  R\s  called  the  resultant  of  Fi,  Fo,  which  are  called 
components  of  R. 

189.  Thus,  the  resultant  R  of  two  forces  Fi,  F2  acting  on 
the  same  particle  is  found  (Fig.  42)  as  the  diagonal  of  the 
parallelogram  constructed  with  Fi.  Fo  as  adjacent  sides. 

142 


190.] 


STATICS  OF   THE   PARTICLE 


143 


Hence 


R  =   VFi-  +  Fa^  +  2FiF2  cos^, 
R 


sin/3 


sina 


sin0 


where  6  is  the  angle  between  Fi  and  i^2,  «  that  between  J? 
and  Fi,  /3  that  between  /^  and  F-y. 

This  proposition  is  known  as  the  parallelogram  of  forces. 
It  enables  us  to  find  the  vector  R  when  the  vectors  F],  F^ 
are  given;  and  conversely,  to  find  Fi,  F2  if,  in  addition  to  the 

'3 


Fig.  42 


vector  R,  the  directions  of  Fi,  Fo  (the  angles  a,  /3)  are  given. 
The  latter  operation  is  called  resolving  a  force  along  given 
directions. 

To  find  7^  when  Fi,  F2  are  given  it  suffices  (instead  of  con- 
structing the  whole  parallelogram)  to  lay  off  (Fig.  43)  1  2, 
equal  to  the  vector  Fi  (in  magnitude,  direction,  and  sense), 
and  2  3,  equal  to  the  vector  F2;  then  1  3  is  the  resultant  R. 
123  is  called  the  triangle  of  forces. 

190.  Let  any  number  71  of  forces  Fi,  F2,  •  •  •  F„  be  applied 
at  the  same  point  0,  i.  e.  act  on  the  same  particle  at  0.  By 
Art.  189,  we  can  find  the  resultant  Ri  of  Fi  and  Fo,  next  the 
resultant  7?2  of  7?i  and  F3,  thvn  the  resultant  R3  of  Ro  and 
Fi,  and  so  on.  The  resultant  R  of  Rn--  and  Fn  is  evidently 
equivalent  to  the  whole  system  Fi,  F2,  F3,  •  •  •  F„,  and  is 


144 


STATICS 


[191. 


called  its  resultant.  It  thus  appears  that  a  system  consisting 
of  any  niwiber  of  forces  acting  on  the  same  particle  is  equivalent 
to  a  single  resultant. 

It  may  of  course  happen  that  this  resultant  is  zero.  In 
this  case  the  system  is  said  to  be  in  equilibrium.  The  con- 
dition of  equilibrium  of  a  system  of  forces  acting  on  the  same 

particle  is  therefore: 

R  =  0. 

The  system  of  forces  in  this  case  produces  no  acceleration; 
notice  that  equilibrium  of  the  forces  does  not  mean  that  the 
particle  is  at  rest.  Under  forces  that  are  in  equilibrium  the 
particle,  if  at  rest,  will  remain  at  rest;  if  in  motion,  it  will 
continue  to  move  uniformly  in  a  straight  line. 

191.  In  practice,  the  process  of  finding  the  resultant 
indicated  in  Art.  190  is  inconvenient  when  the  number  of 
forces  is  large.     If  the  forces  are  given  geometrically,  as 


Fig.  44. 


vectors,  we  have  only  to  add  these  vectors;  and  this  can  best 
be  done  in  a  separate  diagram,  called  the  force  polygon. 
Thus,  in  Fig.  44,  1  2  is  drawn  equal  and  parallel  to  Fi,  2  3 
equal  and  parallel  to  F.,  3  4  to  F3,  4  5  to  F^,  5  6  to  F^.     The 


194.]  STATICS   OF   THE   PARTICLE  145 

closing  line  of  the  force  polygon,  viz.  1  6  in  the  figilre,  is 
equal  and  parallel  to  the  resultant  R,  which  is  therefore 
obtained  by  drawing  through  the  point  of  application  of  the 
forces  a  line  equal  and  parallel  to  1  6. 

The  geometrical  condition  of  equilibrium  consists  in  the 
closing  of  the  force  polygon,  that  is,  in  the  coincidence  of  its 
terminal  point  6  with  its  initial  point  1. 

192.  Analytically,  a  system  of  concurrent  forces  is  reduced 
to  its  rnost  simple  equivalent  form,  i.  e.  to  its  single  resultant, 
by  resolving  each  force  F  into  three  components  A^,  Y,  Z, 
along  three  rectangular  axes  passing  through  the  particle,  or 
point  of  application  of  the  given  forces.  All  components 
lying  in  the  direction  of  the  same  axis  can  then  be  added 
algebraically,  and  the  whole  system  of  forces  is  found  to 
be  equivalent  to  three  rectangular  forces  SX,  SF,  SZ,  which, 
by  the  parallelogram  law,  can  be  replaced  by  a  single  resultant 


7^  =   V(2Ap  +  (SF)2  +  (2Z)2. 

The  angles  a,  /S,  7  made  by  this  resultant  with  the  axes 
are  given  by  the  relations 

cosa  _  cosjg  _  C0S7  ^  1^ 
SX  ~   SF   ~    SZ    ~  R' 

193.  If  the  forces  all  lie  in  the  same  plane,  only  two  axes 
are  required  and  we  have 

SF 


R  =   V(2X)2  +  (SF)2,     tan0  =  ^, 

where  6  is  the  angle  between  the  axis  of  X  and  R. 

194.  The  condition  of  equilibrium  (Art.  190)  R  =  0  be- 
comes, by  Art.  192,   {^Xy  +  (SF)^  +  (SZ)^  =  0.      As  all 
terms  in  the  left-hand  member  are  positive,  their  sum  can 
vanish  only  when  each  term  is  zero.     The  analytical  conditions 
11 


146  STATICS  [195. 

of  the  equilibrium  of  any  mmiber  of  concurrent  forces  are 

therefore : 

SX  =  0,     2F  =  0,     SZ  =  0. 

195.  The  forces  of  nature  receive  various  special  names 
according  to  the  circumstances  under  which  they  occur. 
Thus  the  weight  of  a  mass  has  already  been  defined  (Art. 
176),  as  the  force  with  which  the  mass  is  attracted  by  the 
mass  of  tlie  earth. 

A  string  carrying  a  mass  at  one  end  and  suspended  from 
a  fixed  point,  is  subjected  to  a  certain  tension.  This  means 
that  if  the  string  were  cut  it  would  require  the  application 
of  a  force  along  the  line  of  the  string  to  keep  the  weight  in 
equilibrium.  This  force,  which  may  thus  serve  to  replace 
the  action  of  the  string,  is  called  its  tension. 

When  the  surfaces  of  two  physical  bodies  A,  B  are  in 
contact,  a  pressure  may  exist  between  them;  that  is,  if  one 
of  the  bodies,  say  B,  be  removed,  it  may  require  the  intro- 
duction of  a  force  to  keep  A  in  the  same  state  of  rest  or 
motion  that  it  had  before  the  removal  of  B.  This  force, 
which  acts  along  the  common  normal  of  the  surfaces  at  the 
point  of  contact  if  there  is  no  friction,  is  called  the  resistance, 
or  reaction,  of  B,  and  a  force  equal  and  opposite  to  it  is 
called  the  pressure  exerted  by  A  on  B.  For  the  case  of 
friction  see  Arts.  237  sq. 

[196.  Exercises. 

(1)  Show  that  the  resultant  of  two  equal  forces  F  including  an  angle 
6  is  2F  cos^O.  Observe  the  variation  of  the  resultant  as  9  varies  from 
0  to  tt;  for  what  angle  e  is  the  resultant  equal  to  F? 

(2)  Show  that  the  resultant  of  two  forces  OA,  OB  is  twice  OC, 
where  C  is  the  midpoint  of  A  and  B. 

(3)  Find  the  magnitude  and  direction  of  the  resultant  of  two  forces 
of  12  and  20  lb.,  including  an  angle  of  60°. 


196.]  STATICS   OF  THE   PARTICLE  147 

(4)  Find  the  resultant  of  6  equal  concurrent  forces,  each  inclined 
to  the  next  at  45°. 

(5)  Show  that  the  forces  OA,  OB,  OC  are  in  equilibrium  if  0  is  the 
centroid  of  the  triangular  area  ABC. 

(6)  Show  (by  Art.  194)  that  if  any  number  of  concurrent  forces  are 
in  equilibrium,  their  point  of  concurrence  is  the  centroid  of  their  ex- 
tremities. 

(7)  A  mass  m  rests  on  a  plane  inclined  to  the  horizon  at  an  angle 
6;  it  is  kept  in  equilibrium:  (a)  by  a  force  Pi  parallel  to  the  plane; 
(b)  by  a  horizontal  force  Pi]  (c)  by  a  force  P?  inclined  to  the  horizon 
at  an  angle  6  +  a.  Determine  in  each  case  the  force  P  and  the  pres- 
sure R  on  the  plane. 

(8)  A  weight  W  is  suspended  from  two  fixed  points  A,  B  hy  means 
of  a  string  AC B,  C  being  the  point  of  the  string  where  the  weight  W 
is  attached.  If  AC,  BC  be  inclined  to  the  vertical  at  angles  a,  §,  find 
the  tensions  in  AC,  BC:  (a)  analytically;  {h)  graphically. 

(9)  Show  that  the  resultant  R  of  three  concurrent  forces  A,  B,  C  in 
the  same  plane  is  given  by  P^  =  ^2  _[_  52  ^  (72  _(_  2BC  cos{B,  C)  + 
2CA  cos(C,  A)  +  2AB  cos(^,  B). 

(10)  A  weightless  rod  AC,  hinged  at  one  end  A  so  as  to  be  free  to 
turn  in  a  vertical  plane,  is  held  in  a  horizontal  position  by  means  of  the 
chain  BC,  the  point  B  lying  vertically  above  A.  If  a  weight  W  be 
suspended  at  C,  find  the  thrust  P  in  ^C  and  the  tension  T  of  the  chain. 
Assume  AC  =  8  ft.,  AB  =  Q  ft. 

(11)  In  Ex.  (10),  suppose  the  rod  AC,  instead  of  being  hinged  at 
A,  to  be  set  firmly  into  the  wall  in  a  horizontal  position;  and  let  the 
chain  fastened  at  B  run  at  C  over  a  smooth  pulley  and  carry  the  weight 
W.  Find  the  tension  of  the  chain  and  the  magnitude  and  direction 
of  the  pressure  on  the  pulley  at  C. 

(12)  In  "tacking  against  the  wind,"  let  W  be  the  force  of  the  wind; 
a,  ^  the  angles  made  by  the  axis  of  the  boat  with  the  direction  in  which 
the  wind  blows,  and  with  the  sail,  respectively.  Determine  the  force 
that  drives  the  boat  forward  and  find  for  \\'hat  position  of  the  sail  it 
is  greatest. 

(1.3)  A  cylinder  of  weight  W  rests  on  two  inclined  planes  whose  inter- 
section is  horizontal  and  parallel  to  the  axis  of  the  cylinder.  Find  the 
pressures  on  these  planes. 


148  STATICS  1196. 

(14)  Find  the  tensions  in  the  string  ABCD,  fixed  at  A  and  D,  and 
carrying  equal  weights  W  at  B  and  C,  ii  AD  =  c  is  horizontal,  AB  — 
BC  =  CD,  and  the  length  of  the  string  is  3L 

(15)  In  the  toggle-joint  press  two  equal  rods  CA,  CB  are  hinged 
at  C]  a  force  F  bisecting  the  angle  2a  between  the  rods  forces  the 
ends  A,  B  apart.  If  A  be  fixed,  find  the  pressure  exerted  at  B  at 
right  angles  to  F  ii  F  =  100  lbs.  and  a  =  15°,  30°,  45°,  60°,  75°,  90°. 

(16)  A  stone  weighing  800  lbs.  hangs  from  a  derrick  by  a  chain  15 
ft,  long.  If  pulled  5  ft.  away  from  the  vertical  by  means  of  a  hori- 
zontal rope  attached  to  it,  what  are  the  tensions  of  the  chain  and  the 
rope?     What  if  pulled  9  ft.  away? 

(17)  A  rope  16  ft.  long  has  its  ends  fastened  to  two  points,  10  ft. 
apart,  at  the  same  height  above  the  ground;  a  weight  W  is  suspended 
from  the  rope  by  means  of  a  ring  free  to  slide  along  the  rope.  Find  the 
tension  of  the  rope. 

(18)  A  string  with  equal  weights  W  attached  to  its  ends  is  hung 
over  two  smooth  pegs  A,  B  fixed  in  a  vertical  wall.  Find  the  pressure 
on  the  pegs:  (o)  when  the  line  AB  is  horizontal;  (b)  when  it  is  inclined 
to  the  horizon  at  an  angle  d. 


CHAPTER  XI. 
STATICS  OF  THE  RIGID  BODY. 

197.  A  system  of  forces  acting  on  a  rigid  body  can,  in 
general,  not  be  reduced  to  a  single  resultant,  as  is  the  case 
for  concurrent  forces  (Art.  190) ;  in  other  words,  there  does 
not  always  exist  a  single  force  having  the  same  effect  that 
the  system  of  forces  has  in  changing  the  motion  of  the  body. 

Before  discussing  the  general  case  it  is  best  to  consider 
certain  particular  kinds  of  systems  of  forces,  viz.  concurrent, 
parallel,  and  complanar  systems. 

Throughout  the  statics  of  the  rigid  body  it  is  assumed  that 
the  effect  of  a  force  is  not  changed  if  the  force  is  transferred  to 
any  other  position  on  its  line  of  action;  in  other  words,  a  body 
is  called  rigid  if,  and  only  if,  it  possesses  this  property. 
Thus  the  ''point  of  application"  of  a  force  acting  on  a  rigid 
body  is  not  an  essential  characteristic  of  the  force;  what 
characterizes  the  force  is  its  magnitude,  line  of  action,  and 
sense.  This  is  what  is  meant  by  saying  that  a  force  is  a 
localized  vector  or  rotor  (Art.  182). 

1.  Concurrent  forces. 

1Q8.  In  the  case  of  concurrent  forces  there  exists  a  single 
resultant,  viz.  the  geometric  sum  of  the  forces.  If  this 
resultant  happens  to  be  zero,  i.  e.  if  the  force  polygon  (Art. 
191)  closes,  the  forces  are  in  equilibrium. 

As  the  projection  of  a  closed  polygon  on  any  line  is  zero, 
it  follows  that  the  projection  of  the  resultant  on  any  liiie  is 
equal  to  the  algebraic  sum  of  the  projections  of  its  components. 

149 


150 


STATICS 


[199. 


Thus,  if  the  forces  P,  Q  intersect  at  0  and  have  the  re- 
sultant R  we  find  by  projecting  on  any  line  I: 

R  cos(?,  R)  =  P'cosil,  P)  -\-Q  cos{l,  Q). 

Let  the  hne  I  be  drawn  through  0,  in  the  plane  of  P  and 
Q,  and  let  an  arbitrary  length  OS  =  s  (Fig.  45)  be  laid  off 

at  right  angles  to  /  in  the  same 
plane.  Then,  multiplying  the 
last  equation  by  .s  we  find 

R-scos(l,R)  =  P-scos(/,  P)  + 
Q-scos(l,Q); 

or  since  s  cos(Z,  R)  =  r, 
s  cos{l,  P)  =p,  s  cos{l,  Q)  ^  q 
are  the  perpendiculars  from  S 
to  R,P,Q: 

Rr  =  Pp  +  Qq. 

Now  the  product  of  a  force  into  its  perpendicular  distance 
from  a  point  is  called  the  moment  of  the  force  about  the 
point;  the  product  is  taken  with  the  positive  or  negative 
sign  according  as  the  force  tends  to  turn  counterclockwise 
or  clockwise  about  the  point.  We  have  therefore  proved 
that  the  algebraic  sum  of  the  moments  of  any  two  intersect- 
ing forces  about  anij  point  in  their  plane  is  equal  to  the  moment 
of  their  resultant  about  the  same  point. 

This  proposition  is  known  as  the  theorem  of  moments, 
or  Varignon's  theorem.  It  is  readily  extended  to  any 
number  of  concurrent  forces  in  the  same  plane.  As  a  corollary 
it  follows  that  the  sum  of  the  moments  of  any  such  forces 
about  any  point  of  their  resultant  is  zero. 

199.  As  the  moment  of  a  force  represents  twice  the  area 
of  the  triangle  having  the  force  as  base  and  the  reference 


Fig.  45. 


200.] 


STATICS  OF  THE   RIGID   BODY 


151 


point  as  vertex,  the  theorem  of  moments  can  also  be  proved 

by  comparing  areas.     Thus,  with  the  notation  of  Fig.  46 

we  have 

SOR  =  SOQ  +  SQR  +  QOR, 


I.  e. 


or  smce 


->R 


R.r  =  Q-q+  P-ST+  PTU, 
ST  +  TU  =  SU  =  p: 

Rr  =  Qq  +  Pp. 

It  is  often  convenient  to  think  of  the  moment  Rr  of  a 
force  R  about  the  point  S  as  a  vector  drawn  through  S  at 
right  angles  to  the  plane  deter- 
mined by  S  and  R.  This  is  in 
agreement  with  the  representa- 
tion of  a  parallelogram  area  by 
such  a  vector,  mentioned  in  Art. 
119.  Indeed,  the  moment  Rr  is 
the  cross-product  of  the  radius 
vector  drawn  from  S  to  any  point 
of  R  into  tlie  force-vector  R. 

This  representation  is  of  special  advantage  when  the 
concurrent  forces  do  not  lie  in  the  same  plane.  It  can  then 
be  shown  that  the  moment  of  the  resultant  about  any  point 
is  equal  to  the  geometric  sum  of  the  vectors  representing 
the  moments  of  the  components. 

2.  Parallel  forces. 
200.  It  will  be  proved  in  the  next  article  that  any  two 
parallel  forces  acting  on  a  rigid  body  have  a  single  resultant, 
except  when  the  two  parallel  forces  are  of  equal  magnitude 
and  opposite  sense.  In  the  latter  case,  the  two  equal  and 
opposite  parallel  forces  are  said  to  constitute  a  couple, 
and  no  further  reduction  is  possible. 


FiR.  46. 


152 


STATICS 


[201. 


It  follows  readily  that  any  system  of  parallel  forces  acting 
on  a  rigid  body  can  be  reduced  either  to  a  single  force  or  to  a 
single  couple. 

201.  Resultant  of  two  parallel  forces.  In  the  plane  of 
the  given  parallel  forces  P,  Q,  resolve  P,  at  any  point  p  of 
its  line  of  action,  into  any  two  components,  say  P'  and  F 
(Fig.  47);  and  at  the  point  q  where  F  meets  the  line  of  Q, 


Fig.  47. 


resolve  Q  into  two  components  F',  Q',  selecting  for  F'  a 
force  equal  and  opposite  to,  and  in  the  same  line  with,  F. 
The  two  equal  and  opposite  forces  F,  F'  in  the  same  line 
pq  have  no  effect  on  the  rigid  body  so  that  the  given  forces 
P,  Q  are  together  equivalent  to  the  two  components  P', 
Q'  alone.  The  lines  of  P'  and  Q'  will  in  general  intersect 
at  a  point  r  and  these  forces  can  therefore  be  replaced  by 
a  resultant  R  passing  through  r. 

By  placing  the  triangles  pP'P  and  qF'Q  together  so  that 
their  equal  sides  PP'  and  qF'  coincide  (as  is  done  in  Fig.  47, 
on  the  right)  it  appears  at  once  that  the  resultant  of  P'  and 


202]  STATICS  OF  THE  RIGID  BODY  153 

Q' ,  and  hence  the  resultant  R  of  P  and  Q,  is  parallel  to  P  and 
Q  and  in  magnitude  equal  to  the  algebraic  sum  of  P  and  Q: 

R^  P  +  Q. 

In  Fig.  47,  the  two  given  parallel  forces  P,  Q  were  as- 
sumed of  the  same  sense.  The  construction  applies,  how- 
ever, equally  well  to  the  case  when  they  are  of  opposite  sense. 
The  resultant  R  will  then  be  found  to  lie  not  between  P  and 
Q,  but  outside,  on  the  side  of  the  larger  force.  The  con- 
struction fails  only  when  the  two  given  forces  are  equal  and 
of  opposite  sense,  since  then  the  lines  pP'  and  qQ'  become 
parallel.     This  exceptional  case  will  be  considered  in  Art.  208. 

202.  The  theorem  of  moments  for  parallel  forces.  As  the 
forces  R,  P',  Q'  (Fig.  47)  are  concurrent  the  theorem  of 
moments  (Art.  198)  can  be  applied  to  these  three  forces. 
Hence,  taking  moments  about  any  point  S  of  the  plane  of 
P'  and  Q'  and  denoting  the  perpendiculars  from  S  to  the 
forces  by  the  corresponding  small  letters,  we  have* 
Rr  =  P'p'  +  Q'q'. 

Now  P'  can  be  regarded  as  the  resultant  of  P  and  —  F,  and 
Q'  as  the  resultant  of  Q  and  —  F' ;  hence 

P'p'  =  Pp  -  Ff,     Q'q'  =Qq-  F'f; 

substituting  these  values  and  remembering  that  F  and  F'  are 
equal  and  opposite  and  in  the  same  line,  we  find 
Rr  =  Pp  +  Qq; 

i.  e.  the  sum  of  the  moments  of  two  parallel  forxes  about  any 
point  in  their  plane  is  equal  to  the  moment  of  their  resultant 
about  the  sanUb  point. 

If,  in  particular,  the  point  of  reference  be  taken  on  the 
resultant  so  that  r  =  0,  we  find 

Pp  =  -  Qq; 


154  •  STATICS  [203. 

i.  e.  the  resultant  of  two  'parallel  forces  divides  their  distance 
in  the  inverse  ratio  of  the  forces. 

This  proposition,  well  known  from  its  application  to  the 
lever,  is  often  referred  to  as  the  principle  of  the  lever. 

203.  It  has  been  shown  that  two  parallel  forces  P,  Q  acting 
on  a  rigid  body,  provided  they  are  not  equal  and  of  opposite 
sense,  have  a  resultant  R  =  P  -\-  Q,  parallel  to  P  and  Q,  and 
that  its  position  in  the  rigid  body  can  be  found  either  ana- 
lytically from  the  fact  that  R  divides  the  distance  between  P 
and  Q  in  the  inverse  ratio  of  these  forces,  or  geometrically 
by  the  construction  of  Art.  201. 

This  geornetrical  construction  is  best  carried  out  in  the 
tollowing  order  (Fig.  48).     The  parallel  forces  P,  Q  being 


._->0 


Fig.  48. 

given  in  position,  begin  by  constructing  the  force  polygon, 
which  here  consists  merely  of  a  straight  line  on  which  the 
forces  P  =  12,  Q  =  23  are  laid  off  to  scale ;  the  closing  line, 
1  3,  gives  the  resultant  in  magnitude,  direction,  and  sense; 
it  only  remains  to  find  its  position,  and  for  this  it  suffices  to 
find  one  point  of  its  line  of  action. 

Now,    to    resolve    P   and   Q   each    into  two  components 
(as  is   done  in  Art.  201)  so  that  one  component  of  P  and 


205.]  STATICS   OF  THE   RIGID   BODY  155 

one  of  Q  are  equal  and  opposite  and  in  the  same  line,  it  is 
only  necessary  to  draw  from  an  arbitrary  point  0,  called  the 
pole,  the  lines  0  1,  0  2,  0  3;  then  1  0,  0  2  can  be  regarded 
as  components  of  P  =  1  2,  and  2  0,  0  3  as  components  of 
Q  =  23.  Next  construct  the  so-called  funicular  polygon  by 
drawing  a  hne  I  parallel  to  0  1,  intersecting  P  say  at  p; 
through  p  a  line  II  parallel  to  0  2  meeting  Q  say  at  q;  through 
q  a  line  III  parallel  to  0  3. 

The  intersection  r  of  I  and  III  is  a  point  of  the  resultant  R 
as  appears  by  comparing  Figs.  48  and  47;  Fig.  48  being  the 
same  as  Fig.  47,  with  the  superfluous  lines  left  out. 

204.  Analytically,  the  resultant  of  n  parallel  forces  Fi, 
F2,  •  •  •  Fn,  whether  in  the  same  plane  or  not,  can  be  found  as 
follows : 

The  resultant  of  Fi  and  F2  is  a  force  Fi  +  F2  situated  in 
the  plane  (Fi,  F2),  so  that  F^pi  =  F2P2  (Art.  202),  where 
Pi,  P2  are  the  (perpendicular  or  oblique)  distances  of  the 
resultant  from  Fi  and  Fo,  respectively.  This  resultant  Fi 
+  F2  can  now  be  combined  with  Fz  to  form  a  resultant 
Fi-\-  F2  +  Fz,  whose  distances  from  Fi  +  F2  and  F3  in  the 
plane  determined  l^y  these  two  forces  are  as  Fz  is  to  Fi  +  F2. 
This  process  can  be  continued  until  all  forces  have  been 
combined;  the  final  resultant  is 

Pi   +   7^2   +    •  •  •    +   Pn. 

Amj  number  of  parallel  forces  are,  therefore,  equivalent  to  a 
single  resultant  equal  to  their  algebraic  sum,  provided  this  sum 
does  not  vanish. 

205.  To  find  the  position  of  this  resultant  analytically,  let 
the  points  of  application  of  the  forces  Pi,  P2,  •  •  •  Fn  be 
(xi,  yi,  Zi),  (x2,  y2,  Z2),  •  •  •  (xn,  Vn,  Zn)  ■  The  point  of  applica- 
tion of  the  resultant  Pi  -f  Po  of  Pi  and  P2  may  be  taken  so  as 


156  STATICS  [206. 

to  divide  the  distance  of  the  points  of  application  of  Fi  and 
F<2,  in  the  ratio  F^jFi;  hence,  denoting  its  co-ordinates  by 

x',  y',  z' ,  we  have  Fi{x'  —  Xi)  =  F^ix^  —  x'),  or 

(Fi  +  F~^x'  =  F,x,  +  F2X2, 

and  similarly  for  7/  and  z\ 

The  force  Fi  +  F2  coml)ines  with  F3  to  form  a  resultant 
F1  +  F2+F3,  whose  point  of  apphcation  {x" ,y" ,  z")  is  given  Iw 

(Fi  +  F2  +  F3)x"  =  Fixi  +  F2X2  +  Fszs, 

with  similar  expressions  for  y",  z" . 

Proceeding  in  this  way,  we  find  for  the  point  of  application 
{x,  y,  z)  of  the  resultant  of  all  the  given  forces 

(Fi  +  F2  +  •  •  •  +  Fn)x  =  F,xr  +  F,Xo  +  •  •  •  +  F„a'„, 

with  corresponding  equations  for  y  and  I.     We  may  write 
these  equations  in  the  form: 

-  _  ^^      -  _  ^l]L      -  -  ^ 
^  ~    2F '     ^  ~    SF  '     ^  ~  ZF  ' 
unless  2F  =  0. 

As  these  expressions  for  x,  y,  z  are  independent  of  the 
direction  of  the  parallel  forces  it  follows  that  the  same  point 
(x,  y,  z)  would  be  found  if  the  forces  were  all  turned  in  any 
way  about  their  points  of  application,  provided  they  remain 
parallel.  The  point  {x,  y,  z)  is  for  this  reason  called  the 
center  of  the  system  of  parallel  forces.  It  is  nothing  but  the 
centroid  of  the  points  of  application  if  these  points  are  re- 
garded as  possessing  masses  equal  to  the  magnitudes  of  the 
forces. 

206.  Conditions  of  equilibrium.  It  follows  from  what  pre- 
cedes that  Jor  the  equilibrium  of  a  system,  of  'parallel  forces  the 
condition  2F  —  Q,  or  R  =  Q,  though  always  necessary,  is  not 
sufficient. 


207.]  STATICS  OF  THE  RIGID  BODY  157 

Now,  if  the  resultant  R  of  the  n  parallel  forces  Fi,  F2,  ■  •  •  F„ 
is  zero,  the  resultant  R'  of  the  n  —  1  forces  Fi,  F2,  •  •  ■  Fn-i 
cannot  be  zero,  and  its  point  of  application  is  found  (by  Art. 
205)  from  x  =  (F^Xi  +  F2X2 -\-  •  •  •  +  F n-iXn-i)  I {F ,  -\- F^  + 
•  •  •  Fn-i)  and  similar  expressions  for  y  and  z.  The  whole 
system  of  parallel  forces  is  therefore  equivalent  to  tl.e  two 
parallel  forces  R'  and  Fn-  Two  such  forces  can  be  in  equi- 
librium only  when  they  lie  in  the  same  straight  line;  i.  e.  Fn 
must  lie  in  the  same  line  with  R'  and  must  therefore  pass 
through  the  point  (x,  y,  z),  which  is  a  point  of  R'. 

The  additional  condition  of  equilibrium  is,  therefore, 

X       Xn        y        yn       Z       Zn 


cosa  cos/3  C0S7   ' 

where  a,  jS,  7  are  the  angles  made  by  the  direction  of  the 

forces  with  the  axes. 

For  practical  application  it  is  usually  best  to  replace  the 

last  condition  l)y  taking  moments  about  a  convenient  point. 

Thus,  the  analytical  conditions  of  equilibrium  can  be  written 

in  the  form 

SF  =  0,     Si^p  =  0. 

Graphically,  to  the  former  corresponds  the  closing  of  the 
force-polygon,  to  the  latter,  in  the  case  of  complanar  forces, 
the  closing  of  the  funicular  polygon. 

207.  Weight;  center  of  gravity.  The  most  important 
special  case  of  parallel  forces  is  that  of  the  force  of  gravity 
which  acts  at  any  given  place  near  the  earth's  surface  in 
approximately  parallel  lines  on  every  particle  of  matter. 

If  g  be  the  acceleration  of  gravity,  the  force  of  gravity  on  a 
particle  of  mass  m  is 

w  =  mg, 

and  is  called  the  weight  of  the  particle  or  of  the  mass  m. 


158  STATICS  [208. 

For  a  system  of  particles  of  masses  w-i,  iih,  •  •  •  nin  we  have 

W\  =  '^niQ,     W2  =  rrhg,     •  •  ■  Wn  —  mnQ- 
If  the  particles  are  rigidly  connected,  the  resultant  W  of 
these  parallel  forces, 

W  =  Wi-\-  102+  ■  ■  ■  +  Wn  =  (wh  +  //?2  +  •  •  •  +  mn)g  =  Mg, 

where  M  is  the  mass  of  the  system,  is  called  the  weight  of  the 
system. 

The  center  of  the  parallel  forces  of  gravity  of  a  system  of 
rigidly  connected  particles  has,  by  Art.  205,  the  co-ordinates 

'  _  _  Xmgx      _  _  ^mgy      _  _  If^nigz 
~   '^mg  '     ^  "    :^???g  '        "   2/?ig  ' 

or  since  the  constant  g  cancels, 

_  _  1,mx       _  _  Zmy      _  _  llmz 
^^^m^'     ^~~^i'     ^~S^' 

This  point  is  called  the  center  of  gravity  of  the  system, 
and  is  evidently  identical  with  the  center  of  mass,  or  centroid 
(see  Art.  159). 

For  continuous  masses  the  same  formulse  hold,  except  that 
the  summations  become  integrations. 

The  weight  TF  of  a  physical  body  of  mass  M  is  therefore  a 
vertical  force  passing  through  the  centroid  of  its  mass. 

3.  Theory  of  couples. 

208.  The  construction  given  for  the  resultant  of  two  par- 
allel forces  given  in  Arts.  201  and  203  fails  if,  and  only  if,  the 
given  forces  are  equal  and  of  opposite  sense.  In  this  case, 
the  lines  pP'  and  qQ'  in  Fig.  47,  and  the  lines  I  and  III  of  the 
funicular  polygon  (Fig.  48),  become  parallel,  so  that  their 
intersection  r  lies  at  infinity.  The  magnitude  of  the  resultant 
is  of  course  zero. 


208.] 


STATICS   OF   THE   RIGID   BODY 


159 


The  combination  of  two  equal  and  opposite  parallel  forces 
{F,  —  F)  acting  on  a  rigid  body  is  called  a  couple.  A  couple 
is,  therefore,  not  equivalent  to  a  single  force,  although  it  might 
be  said  to  be  equivalent  to  the  limit  of  a  force  whose  mag- 
nitude approaches  zero  while  its  line  of  action  is  removed 
to  infinity. 

The  perpendicular  distance  AB  =  p  (Fig.  49)  of  the  forces 
of  the  couple  is  called  the  arm,  and  the  product  Fp  of  the 
force  F  into  the  arm  p  is 
called  the  moment  of  the 
couple.  The  moment,  or  B 
the  couple  itself,  is  also  — F 
called  a  torque. 

Notice  that  the  moment 
of  a  couple  is  simply  the 
sum  of  the  moments  of  its 
forces  about  any  point  in 
its  plane. 

If  we  imagine  the 
couple  {F,  p)  to  act  upon  an  invariable  plane  figure  in  its 
plane,  and  if  the  midpoint  of  its  arm  be  a  fixed  point  of 
this  figure,  the  couple  will  evidently  tend  to  turn  the 
figure  about  this  midpoint.  (It  is  to  be  observed  that  it 
is  not  true,  in  general,  that  a  couple  acting  on  a  rigid  body 
produces  rotation  about  an  axis  at  right  angles  to  its  plane.) 
A  couple  of  the  type  {F,  p)  or  {F' ,  p')  (see  Fig.  49)  will  tend 
to  rotate  counterclockwise,  while  a  couple  of  the  type  {F" ,  p") 
tends  to  turn  clockwise.  Couples  in  the  same  plane,  or 
in  parallel  planes,  are  therefore  distinguished  as  to  their  sense 
and  this  sense  is  expressed  by  the  algebraic  sign  attributed 
to  the  moment.  Thus,  the  moment  of  the  couple  {F,  p)  in 
Fig.  49  is  +  Fp,  that  of  the   couple  {F",  p")  is  -  F"p", 


Fis.  49. 


160 


STATICS 


[209. 


-F 


Fig.  50. 


209.  The  effect  of  a  couple  is  not  changed  by  translation,  i.  c.  by 
moving  its  plane  parallel  to  itself  without  rotating  it. 

Let  AB  =  p  (Fig.  50)  be  the  arm 
of  the  couple  {F,  p)  in  its  original 
position,  and  A'B'  the  same  arm  in  a 
new  position  i:)arallel  to  the  original 
one  in  the  same  plane,  or  in  any  par- 
allel plane.  By  introducing  at  each 
end  of  the  new  arm  A'B'  two  oppo- 
site forces  F,  —  F,  each  equal  and 
parallel  to  the  original  forces  F,  the 
given  S3'stem  is  not  changed.  But 
the  two  equal  and  parallel  forces  F 
at  A  and  B'  form  a  resultant  2F  at  the  midpoint  0  of  the  diagonal 
AB'  of  the  parallelogram  ABB' A'.  Similarly,  the  two  forces  —  F  at 
B  and  A'  are  together  equivalent  to  a  resultant  —  2F  at  the  same  point 
O.  These  two  resultant  forces,  being  equal  and  opposite  and  acting  in 
the  same  line,  are  together  equivalent  to  zero.  Hence  the  whole  sys- 
tem reduces  to  the  force  F  at 
A'  and  the  force  —  i^  at  B' , 
which  form,  therefore,  a 
couple  equivalent  to  the  orig- 
inal couple  at  AB. 

210.  The  effect  of  a  couple 
is  not  changed  by  rotation  in 
its  plane. 

Let  AB  (Fig  51)  be  the 
arm  of  the  couple  in  the  orig- 
inal position,  C  its  midpoint, 
and  let  the  arm  be  turned 
about  C  into  the  position 
A'B'.  Applying  again  at  A', 
B'  equal  and  opposite  forces 

each  equal  to  F,  the  forces  -  F  aAA'  and  F  at  A  will  form  a  resultant 
acting  along  CD,  while  F  at  B'  and  -  F  &i  B  give  an  equal  and  oppo- 
site resultant  along  CE.  These  two  resultant  forces  destroy  each  other 
and  leave  nothing  but  the  couple  formed  by  Fat  yl'and  -  F  at  B' 
which  is  therefore  equivalent  to  the  original  couple, 


212.]  STATICS   OF  THE  RIGID   BODY  161 

Any  other  displacement  of  the  couple  m  its  plane,  or  to  a  parallel 
plane,  can  be  effected  by  a  translation  combined  with  a  rotation  in 
its  plane  about  the  midpoint  of  its  arm.  The  effect  of  a  couple  is  therefore 
not  changed  by  any  displacement  in  its  plane  or  to  a  parallel  plane. 

211.  The    effect  of  a   couple    is    not 
changed   if    its  force   F  and    its  arm  j) 
he  changed  simultaneously  in  any  ivay,     p' 
provided    their  product  Fp  remain  the 
same.  ^ 

Let   AB  =  p)   be  the  original  arm         ' 
(Fig.  52),  F  the  original  force  of  the 


B  C 


-F 


F 
A' 

-F' 


couple;  and  let  A'B'  —  p'  be  the  new 

arm.     The  introduction  of  two  equal  '        -p-      rn 

and  opposite  forces  F'  at  A' ,  and  also 

at  B',  will  not  change  the  given  system  F,   —  F.     Now,  selecting  for 

F'  a  magnitude  such  that  F'jj'  =  Fp,  the  force  F  at  A  and  the  force 

-  F'  at  A'  combine  (Arts.  201,  203)  to  form  a  parallel  resultant 
through  C,  the  midpoint  of  the  arm,  since  for  this  point  F  ■  ^p  + 
{-  F')  ■  Ip'  =  0.  Similarly,  -  F  at  B  and  F'  at  B'  give  a  resultant  of 
the  same  magnitude,  in  the  same  line  through  C,  but  of  opposite  sense. 
These  two  resultant  forces  thus  destroying  each  other,  there  remains 
only  the  couple  formed  by  F'  at  A'  and  —  F'  at  B',  for  which 
Fp  =  F'p'. 

212.  It  results  from  the  last  three  articles  that  the  only  essen- 
tial characteristics  of  a  couple  are:  (a)  the  numerical  value  of  the 
moment;  (6)  the  sense,  or  direction  of  rotation;  and  (c)  what  has  been 
called  the  "aspect"  of  its  plane,  i.  e.  the  direction  of  any  normal  to 
this  plane. 

It  is  to  be  noticed  that  the  plane  of  the  two  forces  forming  the 
couple  is  not  an  essential  characteristic  of  the  couple;  just  as  the 
point  of  application  of  a  force  is  not  an  essential  characteristic  of  the 
force  (see  Art.  197);  provided,  of  course,  that  the  couple  (or  force)  is 
acting  on  a  rigid  body. 

Now  the  three  characteristics  enumerated  above  can  all  be  indi- 
cated by  a  vector  which  can  therefore  serve  as  the  geometrical  repre- 
sentative of  the  couple.     Thus,  the  couple  formed  by  the  forces  F, 

—  F  (Fig.  53),  whose  perpendicular  distance  is  p,  is  represented  by 
the  vector  AB  =  Fp  laid  off  on  any  normal  to  the  plane  of  the  couple, 

12 


162 


STATICS 


[213. 


The  sense  is  indicated  by  drawing  the  vector  toward  that  side  of  the 
plane  from  which  the  couple  is  seen  to  rotate  counterclockwise. 

We  shall  call  this  geometrical  representative  AB  oi  the  couple  simply 
the  vector  of  the  couple.  It  is  sometimes  called  its  niomcnl,  or  its  axis, 
or  its  axial  moment. 

213.  As  was  pointed  out  in  Art.  208,  a  couple  can  be  regarded  as 
the  limit  of  a  force  whose  magnitude  approaches  zero  while  its  line  of 
action  is  removed  to  infinity.  Similarly,  in  kinematics  an  angular 
velocity  whose  magnitude  tends  to  zero  while  its  axis  is  removed  in- 
definitely becomes  in  the  limit  a  velocity  of  translation. 


Fig.  .53. 


Just  as,  in  kinematics  (see  Art.  122),  two  equal  and  opposite  angular 
velocities  about  parallel  axes  produce  a  velocity  of  translation,  so  in 
statics  two  equal  and  opposite  forces  along  parallel  lines  form  a  new 
kind  of  quantity  called  a  couple. 

It  should,  however,  be  noticed  that  while  angular  velocities  and 
forces  are  represented  by  rotors,  i.  e.  by  vectors  confined  to  definite 
lines,  velocities  of  translation  and  couples  have  for  their  geometrical 
representatives  vectors  not  confined  to  particular  lines. 

It  is  due  to  this  analogy  between  the  two  fundamental  conceptions 
that  a  certain  dualism  exists  between  the  theories  of  statics  and  kine- 
matics, so  that  a  large  portion  of  the  theory  of  kinematics  of  a  rigid 
body  might  be  made  directly  available  for  statics  by  simply  substituting 
for  angular  velocity  and  velocity  of  translation  the  corresponding  ideas 
of  force  and  couple. 


215.] 


STATICS   OF  THE  RIGID   BODY 


163 


214.  It  is  easily  seen  how,  by  means  of  Arts.  209-211,  any 
number  of  couples  acting  on  a  rigid  body  can  be  reduced  to  a 
single  resultant  couple.  It  can  also  be  proved  without  much 
difficulty  that  the  vector  of  the  resultant  couple  is  the  geo- 
metric sum  of  the  vectors  of  the  given  couples;  in  other  words, 
vectors  reyreseiiting  couples  acting  on  the  same  rigid  body  are 
combined  by  the  parallelograin  law. 

In  the  particular  case  when  the  couples  all  lie  in  parallel 
planes,  or  in  the  same  plane,  their  vectors  may  be  taken  in 
the  same  line  and   can,  therefore,  be  added    algebraically. 

Generally,  the  resultant  of  any  number  of  couples  is  a  single 
couple   whose    vector   is   the 
geometric  sum  of  the  vectors 
of  the  given  couples. 

Conversely,  a  couple  can 
be  resolved  into  components 
by  resolving  its  vector  into 
components. 

215.  To  combine  a  single 
force  P  with  a  couple  {F,p) 
lying  in  the  same  plane  it 
is  only  necessary  to  place 
the  couple  in  its  plane  in 
such  a  position  (Fig.  54) 
that  one  of  its  forces,  say  —  F,  shall  lie  in  the  same  line  and  in 
opposite  sense  with  the  single  force  P,  and  to  transform  the 
couple  {F,  p)  into  a  couple  (P,  p'),  by  Art.  211,  so  that  Fp  = 
Pp'.  The  original  force  P  and  the  force  —  P  of  the  transformed 
couple  destroying  each  other  at  A,  there  remains  only  the 
other  force  P,  at  A',  of  the  transformed  couple,  that  is,  a 
force  parallel  and  equal  to  the  original  single  force  P,  at 
the  distance 


i 

i 

, 

p 

P 

, 

A 

-F 

^ 

v'       > 

< 

'  ~V~' 

K            > 

> 

, 

-P 

1 

f 

Fis.  54. 


1G4  STATICS  [216. 

from  it. 

Hence,  a  couple  and  a  single  force  in  the  same  plane  are 
together  equivalent  to  a  single  force  equal  and  parallel  to,  and  of 
the  same  sense  with,  the  given  force,  but  at  a  distance  from  it 
which  is  found  by  dividing  the  moment  of  the  couple  by  the 
single  force. 

Conversely,  a  single  force  P  applied  at  a  point  A  of  a  rigid 
body  can  always  be  replaced  by  an  equal  and  parallel  force  P 
of  the  same  sense,  applied  at  any  other  point  A'  of  the  same 
body,  in  combination  with  the  couple  formed  by  P  at  A  and  —  P 
at  A'. 

This  follows  at  once  by  applying  at  A'  two  equal  and 
opposite  forces  each  equal  and  parallel  to  P. 

216.  The  proposition  of  Art.  215  applies  even  when  the 
force  lies  in  a  plane  parallel  to  that  of  the  couple,  since  the 
couple  can  be  transferred  to  any  parallel  plane  without  chang- 
ing its  effect. 

If  the  single  force  intersects  the  plane  of  the  couple,  it 
can  be  resolved  into  two  components,  one  lying  in  the  plane 
of  the  couple,  while  the  other  is  at  right  angles  to  this  plane. 
On  the  former  component  the  couple  has,  according  to  Art. 
215,  the  effect  of  transferring  it  to  a  parallel  line.  We  thus 
obtain  two  non-intersecting,  or  skew,  forces  at  right  angles  to 
each  other. 

Let  P  be  the  given  force,  and  let  it  make  the  angle  a  with 
the  plane  of  the  given  couple,  whose  force  is  F  and  whose  arm 
is  p.  Then  P  sina  is  the  component  at  right  angles  to  the 
plane  of  the  couple,  while  P  cosa  combined  with  the  couple 
whose  moment  is  Fp  is  equivalent  to  a  force  P  cosa  in  the 
plane  of  the  couple;  this  force  P  cosa  is  parallel  to  the  pro- 


218.]  STATICS  OF  THE  RIGID   BODY  165 

jection  of  P  on  the  plane,  and  has  the  distance  Fp/P  cosa 
from  this  projection. 

Hence,  in  the  most  general  case,  the  combination  of  a  single 
force  and  a  couple  can  be  replaced  by  the  combination  of  two 
single  forces  crossing  each  other  {without  meeting)  at  right 
angles;  it  can  be  reduced  to  a  single  force  only  when  the  force 
is  parallel  to  the  plane  of  the  couple. 

4.  Complanar  forces. 

217.  If  the  forces  acting  on  a  rigid  body  all  lie  in  the  same 
plane,  i.  e.  if  the  forces  are  complanar,  the  system  can  be 
reduced  to  a  single  force  and  a  single  couple  by  applying  the 
last  proposition  of  Art.  215.  For,  selecting  an  arbitrary 
point  0  of  the  plane  as  point  of  reference,  we  can  replace 
each  force  F  of  the  system  by  an  equal  force  F  applied  at 
0,  together  with  a  couple  Fp,  whose  arm  p  is  the  perpen- 
dicular from  0  to  the  line  of  action  of  the  given  force  F  at  P. 

We  thus  obtain,  in  the  plane,  a  number  of  concurrent  forces 
at  0  which  are  equivalent  to  a  single  resultant  R,  passing 
through  0  and  equal  to  the  geometric  sum  of  the  given  forces ; 
and  in  addition  a  numl)er  of  couples  in  the  same  plane  which 
give  a  single  resultant  couple,  say  H  =  llFp. 

Notice  that  the  moment  H  of  the  resultant  couple  is 
simply  the  sum  of  the  moments  about  0  of  all  the  given 
forces. 

It  follows  that  the  conditions  of  equilibrium  are: 

7^  =  0,     //  =  0; 

i.  e.  a  system  of  complanar  forces  is  in  equilibrium  if,  and  only 
if,  (a)  its  resultant  is  zero,  and  (h)  the  algebraic  sum  of  the 
moments  of  all  its  forces  is  zero  about  any  point  in  its  plane. 

218.  By  Art.  217,  a  system  of  complanar  forces  reduces, 
for  any  point  of  reference  0  in  its  plane,  to  a  force  R  and  a 


166 


STATICS 


I219. 


couple  H.  But  as  these  lie  in  the  same  plane,  it  follows  from 
the  first  proposition  of  Art.  215  that  they  can  be  reduced  to 
a  single  resultant  R  (unless  R  =  0).  The  distance  r  of  this 
smgle  resultant  from  0  is  such  that  Rr  =  —  H;  i.  e.  r  = 
—  H/R.  The  line  of  action  of  this  single  resultant  is  called 
the  central  axis  of  the  system. 

Thus,  a  system  of  complanar  forces  can  always  be  reduced 
either  to  a  single  force  i2  or  to  a  single  couple  H. 

219.  For  a  purelj^  analytical  reduction  of  a  plane  system  of 
forces  the  system  is  referred  to  rectangular  axes  Ox,  Oy, 
arbitrarily  assumed  in  the  plane  (Fig.  55).     Every  force  F  is 


).Y 


-X 


■^X 


X         / 


-Y 


Fig. 


resolved  at  its  point  of  application  P  (x,  y)  into  two  com- 
ponents X,  Y ,  parallel  to  the  axes,  so  that 

X  =  F  cosa,     Y  =  F  sina, 

a  being  the  angle  made  by  F  with  the  axis  Ox.  At  the  origin 
0  two  equal  and  opposite  forces  X,  —  X  are  applied  along 
Ox,  and  two  equal  and  opposite  forces  Y,  —  Y  along  Oy. 
Thus,  X  at  P  is  equivalent  to  X  at  0  together  with  the 
couple  formed  by  X  at  P  and  —  X  at  0;  the  moment  of 


220.]  STATICS   OF   THE   RIGID   BODY  167 

this  couple  is  evidently  —  yX.  Similarly,  F  at  P  is  re- 
placed by  F  at  0  together  with  a  couple  whose  moment  is  xY. 
The  force  P  at  P  is  therefore  equivalent  to  the  two  forces 
X,  F  at  0  together  with  a  couple  whose  moment  is  xY  —  yX. 
Proceeding  in  the  same  way  with  every  given  force,  we 
obtain  a  number  of  forces  X  along  Ox  whose  algebraic  sum 
we  call  2X,  and  a  number  of  forces  F  along  Oy  which  give 
2F.     These  two  rectangular  forces  form  the  resultant 

n  =   -V(SZ)2  +  (2F)2 

whose  direction  is  given  by 

SF 
tana=^^^, 

where  a  is  the  angle  between  Ox  and  R. 

In  addition  to  this,  we  obtain  a  number  of  couples  xY 
—  yX  whose  algebraic  sum  forms  the  resulting  couple 

H  =  SCtF  -  yX). 

The  whole  system  is  thus  found  equivalent  to  a  resultant 
force  R  together  with  a  resultant  couple  H  in  the  same  plane 
with  R  The  conditions  of  equilihrimn  R  =  0,  H  =  0  (Art. 
217)  can  therefore  be  expressed  analytically  by  the  three 
equations 

SX  =  0,     SF  =  0,     Z(xY  -  yX)  =  0. 

220.  If  R  be  not  zero,  R  and  H  can  be  reduced  to  a  single 
resultant  R'  equal  and  parallel  to  R  at  the  distance  —  H/R 
from  it  (see  Art.  218).  The  equation  of  the  line  of  this  single 
resultant  R',  i.  e.  the  central  axis  of  the  system  of  forces,  is 
found  by  considering  that  it  makes  the  angle  a  with  the  axis 
of  X  and  that  its  distance  from  the  origin  is 

H/R  =  ^{xY  -  7/X)/V(:CX)2+  (SF)2. 


168 


STATICS 


[221 


Hence  its  equation  is 

^SF  -  Tj-SZ  -  2(xY  -  yX)  =  0. 
If  i2  =  0,  the  system  is  equivalent  to  the  couple 

H  =  i:(xY  -  7jX). 

If  H  itself  be  also  zero,  the  system  is  in  equilibrium. 
221.  Exercises. 

(1)  A  homogeneous  straight  rod  AB  =  21  (Fig.  56)  of  iveight  W  rests 
with  one  end  A  on  a  smooth  horizontal  plane  AH,  and  with  the  point 
E{AE  =  e)  on  a  cylindrical  support,  the  axis  of  the  cijlinder  being  at 
right  angles  to  the  vertical  plane  coritaining  the  rod.     Determine  what 


horizontal  force  F  must  be  applied  at  a  given  point  F  of  the  rod  {AF  =  f  >  e) 
to  keep  the  rod  in  equilibrium  irhen  inclined  to  the  horizon  at  an  angle  6. 

The  rod  exerts  a  certain  unknown  pressure  on  each  of  the  supports 
at  A  and  E,  in  the  direction  of  the  normals  to  the  surfaces  of  contact, 
provided  there  be  no  friction,  as  is  here  assumed.  The  supports  may 
therefore  be  imagined  removed  if  forces  A,  E,  equal  and  opposite  to 
these  pressures,  be  introduced;  these  forces  A,  E  are  called  the  reactions 
of  the  supports.  The  rod  itself  is  here  regarded  as  a  straight  line;  its 
weight  W  is  applied  at  its  middle  point  C. 

Taking  A  as  origin  and  AH  as  axis  of  x,  the  resolution  of  the  forces 

gives 

•Z.X  =  F  -  E  sine  =  0,  (1) 


sy  =  A  -w  +  E  cose  =  0. 

Taking  moments  about  A,  we  find 

E    e  -W  -l  cos(9  -  F  ■  f  sine  =  0. 


(2) 


(3) 


221.]  .       STATICS   OF  THE   RIGID   BODY  169 

Eliminating  F  from  (1)  and  (3),  we  have 


hence  from  (2), 
and  finally  from  (1), 


_lcosd_ 
^        e  -f  sin20  ^  ' 

\  e  —  f  sm-0  / 


jj,  _  I  sin^  cos9  ^ 
~  e  -f  sm20 


(2)  A  weightless  rod  AB  of  length  I  can  tm-n  freely  about  one  end 
A  in  a  vertical  plane.  A  weight  W  is  suspended  from  a  point  C  of 
the  rod;  AC  =  c.  A  cord  BD  attached  to  the  end  B  of  the  rod  holds 
it  in  equilibrium  in  a  horizontal  position,  the  angle  ABD  being  a  =  150°. 
Find  the  tension  T  of  the  cord  and  the  resulting  pressure  A  on  the 
hinge  at  A. 

(3)  A  cylinder  of  length  21  and  radius  r  rests  with  the  point  A  of 
the  circumference  of  its  lower  base  on  a  horizontal  plane  and  with  the 
point  B  of  the  circumference  of  its  upper  base  against  a  vertical  wall. 
The  vertical  plane  through  the  axis  of  the  cylinder  contains  the  points 
A,  B  and  is  perpendicular  to  the  intersection  of  the  vertical  wall  with 
the  horizontal  plane.  If  there  be  no  friction  at  A,  B,  what  horizontal 
force  F  applied  at  A  will  keep  the  cylinder  in  equilibrium?  When 
is  this  force  /^  =  0? 

(4)  A  weightless  rod  AB  rests  without  friction  on  two  planes  inclined 
to  the  horizon  at  angles  a,  p,  and  carries  a  weight  W  at  the  point  D. 
The  intersection  C  of  these  planes  is  horizontal  and  normal  to  the 
vertical  plane  through  AB.  Find  the  inclination  e  of  AB  to  the  horizon 
and  the  pressures  at  A  and  B. 

(5)  A  weightless  rod  AB  =  I  can  revolve  in  a  vertical  plane  about 
a  hinge  at  A;  its  other  end  B  leans  against  a  smooth  vertical  wall 
whose  distance  from  A  \s  AD  =  a.  At  the  distance  AC  ==  c  from  A 
a  weight  W  is  suspended.  Find  the  horizontal  thrust  A^  at  A  and  the 
normal  pressures  Ay  and  B  aX  A  and  B. 

(6)  The  same  as  (5)  except  that  at  B  the  rod  rests  on  a  smooth  hori- 
zontal cylinder  whose  axis  is  at  right  angles  to  the  vertical  plane  through 
AB.     In  which  of  the  two  problems  is  the  horizontal  thrust  A   at  A  least? 


170  STATICS  .  [222. 

5.  The  general  system  of  forces. 

222.  To  reduce  any  system  of  forces  acting  on  a  rigid  body 
to  its  most  simple  form  the  same  methods  are  used  as  for 
complanar  forces  (comp.  Art.  217) 

Selecting  as  origin  any  point  0  rigidly  connected  with  the 
body,  let  two  equal  and  opposite  forces  F ,  —  i^  be  applied  at 
0,  for  every  one  of  the  given  forces  F.  The  effect  of  the 
given  system  of  forces  on  the  body  is  not  changed  by  the 
introduction  of  these  forces  at  0.  But  we  may  now  regard 
the  given  force  F  acting  at  its  point  of  application  P  as 
replaced  by  the  equal  and  parallel  force  F  at  0,  in  combination 
with  the  couple  formed  by  the  original  force  F  at  P  and  the 
fcroe  —  i^  at  0.  All  the  forces  of  the  given  system  are  thus 
transferred  to  a  common  point  of  application  0,  and  these 
forces  at  0  can  be  replaced  by  a  single  resultant*  R,  passing 
through  0  and  represented  in  magnitude  and  direction  by  the 
geometric  sum  of  the  forces.  In  addition  to  this  resultant  R, 
we  obtain  as  many  couples  {F,  —  F)  as  there  were  forces 
given;  and  their  resultant  is  found  by  geometrically  adding 
the  vectors  of  the  couples  (Art.  214). 

Thus  the  given  system  of  forces  is  seen  to  be  equivalent  to 
a  resultant  R  in  combination  with  a  couple  whose  vector 
we  shall  call  H;  in  other  words,  it  has  been  proved  that  any 
system  of  forces  acting  on  a  rigid  body  can  be  reduced  to  a 
single  resultant  force  in  combincdion  with  a  single  resultant 
couple. 

It   follows   at    once   that    the   geometrical   conditions  of 

equilibrium  are* 

R  =  0,    H  =  0 

223.  Of  the  two  geometrical  elements  representing  a  general 
system  of  forces,  viz.  the  rotor  R  and  the  vector  H,  the  for- 


224.1  STATICS   OF  THE   RIGID   BODY  171 

mer  being  merely  the  geometric  sum  of  the  forces,  is  inde- 
pendent of  the  point  of  reference  0,  while  the  vector  H  is 
in  general  different  for  different  points  of  reference. 

If  the  elements  R,  H  for  a  point  0  are  given,  those  for 
any  other  point  0'  can  readily  be  found.  It  suffices  to  apply 
at  0'  equal  and  opposite  forces  R  and  —  R.  We  then  have 
R  at  0' ,  and  two  couples,  viz.  the  couple  whose  vector  is  H 
and  the  couple  formed  by  7^  at  0  and  —  RaXO';  the  resultant 
of  the  vectors  of  these  two  couples  is  the  vector  H'  corre- 
sponding to  0'.  Here,  as  well  as  in  the  following  articles, 
it  is  assumed  that  R  ^  0;  when  R  =  Q  the  system  reduces 
to  a  couple,  the  same  whatever  the  point  of  reference. 

If  the  new  point  of  reference  0'  had  been  selected  on 
the  line  I  of  the  original  resultant,  no  new  couple  would  have 
been  introduced,  and  H  would  not  have  been  changed.  But 
whenever  the  new  point  of  reference  0'  is  taken  on  a  line  V 
different  from  I,  the  vector  of  the  resultant  couple  H  is 
changed. 

By  increasing  the  distance  r  between  I  and  V  the  moment 
Rr  of  the  additional  couple  is  increased.  The  effect  of  com- 
bining this  additional  couple  Rr  with  H  is,  in  general,  to 
vary  both  the  magnitude  of  the  resulting  vector  H'  and  the 
angle  ^  it  makes  with  the  direction  of  the  resultant  R.  It 
can  be  shown  that  the  line  V  of  the  new  resultant  can  always 
be  selected  so  as  to  reduce  the  angle  </>  to  zero.  The  line  ?o 
for  which  </>  =  0,  i.  e.  for  which  the  vector  H  of  the  resultant 
couple  is  parallel  to  the  resultant  force  R,  is  called  the  central 
axis  of  the  given  system  of  forces.  We  proceed  to  show  how 
it  can  l)c  found  (comp.  Art.  123). 

224.  Let  the  vector  H  be  resolved  at  0  into  a  component 
Ho  =  H  coscf)   along  I,  and  a  component  Hi  =  H  sin<^,  at 


172 


STATICS 


(224. 


right  angles  to  I  (Fig.  57).  In  the  plane  passing  through  I 
at  right  angles  to  Hi,  it  is  always  possible  to  find  a  line  lo 
parallel  to  Z  at  a  distance  ro  from  I,  such  as  to  make  Rro  = 
-Hi. 

The  line  Zo  so  determined  is  the  central  axis.     For,  if  this 
line  be  taken  as  the  line  cf  the  resultant  R,  the  additional 


Ht- 


Fig.  57. 


Fig.  58. 


couple  Rro  destroys  the  component  Hi,  so  that  the  resulting 
couple  ^0  has  its  vector  parallel  to  R. 

As  the  direction  of  the  vector  H  is  always  changed  in 
passing  from  line  to  line,  there  can  be  but  one  central  axis  for 
a  given  system  of  forces. 

It  appears  from  the  construction  of  the  central  axis  given 
above,  that  the  vector  of  the  resulting  couple  for  this 
axis  Zo  is  Ho  =  H  cos0;  it  is,  therefore,  less  than  for  any 
other  line. 

It  is  mstructive  to  observe  how  the  vector  H  increases  and 
changes  its  direction  as  we  pass  from  the  central  axis  Zo  to 
any  parallel  line  I. 


226.J  STATICS   OF  THE   RIGID   BODY  173 

The  transformation  from  lo  to  I  requires  the  introduction  of 
a  couple  whose  vector  Rro  (Fig.  58)  is  at  right  angles  to  the 
plane  {U,  I)  and  combines  with  Hq  to  form  the  resulting 
couple  H  for  I.  As  the  distance  ro  of  I  from  Zo  is  increased, 
both  the  magnitude  of  H  and  the  angle  0  it  makes  with  I 
increase,  the  angle  4>  approaching  ^-tt  as  ro  becomes  infinite. 

225.  It  is  evident  that  since  Ho  =  H  cos0,  the  product 
RH  COS0  is  a  constant  quantity  for  a  given  system  of  forces. 
It  may  be  called  an  invariant  of  the  system. 

If  the  elements  of  reduction  for  the  central  axis  R,  Ho 
be  given,  those  for  any  parallel  line  I  at  the  distance  ro  from 
the  central  axis  are  determined  by  the  equations 

H^  =  Ho'  +  RW,     tancp  =  ■-- . 

no 

To  sum  up  the  results  of  the  preceding  articles,  it  has  been 
shown  that  any  system  of  forces  acting  on  a  rigid  body  can  be 
reduced,  in  an  infinite  number  of  ways,  to  a  resultant  R  in 
combination  with  a  couple  H.  For  all  these  reductions  the 
magnitude,  direction,  and  sense  of  the  resultant  R  are  the 
same,  but  the  vector  H  of  the  couple  changes  according  to 
the  position  assumed  for  the  line  of  R.  There  is  one,  and 
only  one,  position  of  R,  called  the  central  axis  of  the  system, 
for  which  the  vector  H  is  parallel  to  R  and  has  at  the  same 
time  its  least  value,  Ho]  this  value  Ho  is  equal  to  the  projec- 
tion of  any  other  vector  H  on  the  direction  of  the  resultant  R. 

226.  While,  in  general,  a  system  of  forces  cannot  be  reduced 
to  a  single  resultant,  it  can  always  be  reduced  to  two  non- 
intersecting  forces.  This  easily  follows  by  considering  the 
system  reduced  to  its  resultant  R  and  resulting  couple  H 
for  any  point  0  (Fig.  59).  Let  F,  —  F  be  the  forces,  p  the 
arm  of  the  couple  H,  and  place  this  couple  so  that  one  of  the 


174 


STATICS 


[227. 


forces,  say  —  F,  intersects  R  at  0.  Then,  if  R  and  —  F  be 
replaced  by  their  resultant  F' ,  the  given  system  of  forces  is 
evidently  equivalent  to  the  two  non-intersecting  forces  F,  F' 
(compare  Art.  216). 

The  two  forces  F,  F'  determine  a  tetrahedron  OABC;  and 
it  can  be  shown  that  the  volume  of  this  tetrahedron  is  constant 
and  equal  to  one  sixth  of  the  invariant  of  the  system  (Art.  225). 
The  proof  readily  appears  from  Fig.  59.     The  volume  of  the 


tetrahedron  OABC  is  evidently  one  half  of  the  volume  of  the 
quadrangular  pyramid  whose  vertex  is  C  and  whose  base  is 
the  parallelogram  ODAB.  The  area  of  this  parallelogram  is 
Fp  =  // ;  and  the  altitude  of  the  pyramid  is  =  7^  cos0,  being 
equal  to  the  perpendicular  let  fall  from  the  extremity  of  R 
on  the  plane  of  the  couple;  hence  the  volume  of  the  tetra- 
hedron 

=  IRH  COS0  =  iRHo. 

227.  To  effect  the  reduction  of  a  given  system  of  forces  analyt- 
ically, it  is  usually  best  to  refer  the  forces  F  and  their  points 
of  application  P  to  a  rectangular  system  of  co-ordinates  Ox, 
Oy,  Oz  (Fig.  60).  Let  x,  y,  z  be  the  co-ordinates  of  P  and 
X,  Y,  Z  the  components  of  F  parallel  to  the  axes. 


227.1 


STATICS   OF   THE   RIGID   BODY 


175 


To  transfer  these  components  to  0  and  at  the  same  time 
to  introduce  only  couples  whose  vectors  are  parallel  to  the 
axes,  we  proceed  in  two  steps.  Thus  to  transfer,  say  X,  we 
introduce  at  P' ,  the  foot  of  the  perpendicular  let  fall  from  P 
on  the  plane  zx,  two  equal  and  opposite  forces  X,  —  X;  and 
we  do  the  same  thing  at  0.  Then  the  single  force  X  at  P  is 
replaced  by  the  force  A'  at  0  in  combination  with  the  two 
couples  formed  by  X  at  P,  —  X  at  P' ,  and  X  at  P',  —  X 


Fig.  60. 


at  0.  The  vector  of  the  former  couple  is  parallel  to  Oz, 
its  moment  is  —  yX;  the  negative  sign  being  used  because 
for  a  person  looking  on  the  plane  of  the  couple  from  the 
positive  side  of  the  axis  Oz  the  couple  rotates  clockwise.  The 
vector  of  the  latter  couple  is  parallel  to  Oij,  and  its  moment  is 
zX. 

The  transfer  of  Y  to  the  origin  0  requires  the  introduction 
of  two  couples,  —  zY  having  its  vector  parallel  to  Ox  and  xY 
having  its  vector  parallel  to  Oz. 

Finally,  transferring  Z  to  O,  we  have  to  introduce  the 
couples  —  xZ  with  a  vector  parallel  to  Oij,  and  yZ  with  a 
vector  parallel  to  Ox. 


176  STATICS  [228. 

Thus  each  force  F  is  replaced  by  three  forces,  X,  Y,  Z  along 
the  axes  of  co-ordinates  and  applied  at  0,  in  combination 
with  three  couples  whose  vectors  are  yZ  —  zY  parallel  to 
Ox,  zX  —  xZ  parallel  to  Otj,  xY  —  yX  parallel  to  Oz. 

228.  If  this  be  done  for  every  force  of  the  given  system 

and  the  components  having  the  same  direction  be  added,  the 

system  will  be  found  equivalent  to  the  three  rectangular 

forces 

SX,  ZY,  ZZ, 

applied  at  0,  together  with  the  three  couples 

Z{yZ-zY),     i:{zX-xZ),     Z{xY-yX), 

whose  vectors  are  at  right  angles. 

The  three  forces  can  now  be  replaced  by  a  single  resultant 

R  =   V(2ZF+l2F)2TT2Z)2, 

whose  direction  is  determined  by  the  angles  a,  /3,  7  which  it 
makes  with  the  axes  Ox,  Oy,  Oz: 

SX  ^27  SZ 

COSa  =  -^  ,      COS/S  =  —^,      COS7  =  -^. 

In  the  same  way  the  three  couples  can  be  replaced  by  a 
single  resulting  couple  whose  moment  is 

H  =  V[S(?/Z  -  zY)Y  +  [S(2X  -  xZ)Y  +  [Z{xY  -  yX)]\ 

229.  Since  R~,  as  well  as  H"-,  is  thus  found  as  the  sum  of 
three  squares,  each  of  these  quantities  can  vanish  only  if 
the  three  squares  composing  it  vanish  separately.  The  con- 
ditions of  equilibrium  of  a  rigid  body  (Art.  222)  are  therefore 
expressed  analytically  by  the  following  six  equations: 

2X  =  0,     27  =  0,     2Z  =  0, 

2(2/Z  -  zY)  =  0,     2(zX  -  xZ)  =  0,     2(x7  -  yX)  =  0. 


230.]  STATICS  OP  THE  RIGID   BODY  177 

As  the  system  of  co-ordinates  can  be  selected  arbitrarily,  the 

meaning  of  the  first  three  equations  is  that  the  sum  of  the 

components  of  all  the  forces  along  any  three  lines  not  parallel 

to  the  same  plane  must  vanish.     The  last  three  equations 

express  that  the  sum  of  the  moments  of  all  the  forces  about 

any  three  axes  not  parallel  to  the  same  plane  must  also 
vanish. 

The  moment  of  a  force  about  an  axis  must  be  understood  as 
meaning  the  moment  of  its  projection  on  a  plane  at  right 
angles  to  the  axis  with  respect  to  the  point  of  intersection 
of  the  axis  with  the  plane.  This  definition  is  in  accordance 
with  the  somewhat  vague  notion  of  the  moment  of  a  force  as 
representing  its  "  turning  effect."  For,  if  we  regard  the  force 
as  acting  on  a  rigid  body  with  a  fixed  axis,  the  force  can  be  re- 
solved into  two  components,  one  parallel,  the  other  perpen- 
dicular, to  the  axis;  the  former  component  evidently  does 
not  contribute  to  the  turning  effect  which  is,  therefore, 
measured  l^y  the  moment  of  the  latter  alone. 

230.  The  equations  of  the  central  axis  (Art.  223)  can  be 
found  by  a  transformation  of  co-ordinates. 

Let  the  system  be  reduced  for  any  point  0  to  its  resultant  R, 
whose  rectangular  components  we  denote  by 
A  =  2X,     B  =  ^Y,     C  =  2Z, 

and  to  the  vector  H  of  its  resulting  couple  with  the  com- 
ponents 

L  =  2(7/Z  -  zY),     M  =  2(2X  -  xZ),     N  =  2(a;7  -  yX). 

If  a  point  0'  whose  co-ordinates  are  ^,  rj,  f  be  taken  as  new 
point  of  reference  and  the  co-ordinates  of  any  point  with 
respect  to  parallel  axes  through  0'  be  denoted  by  x',  y',  z', 
we  have  x  =  ^-\-x',y  =  'n-[-y',z  =  ^-\-z'.  Substituting 
these  values,  we  find 
13 


178  STATICS  [231. 

L  =  2[(7,  +  y')Z  -  (r  +  z')  Y]  =  v^Z  -  ^ZY  +  :c(^'Z  -  z'Y) 
=  r,C  -  ^B^  L', 

where  L'  is  the  .r-component  of  the  couple  H'  resulting  for 
0'  as  point  of  reference.  Similar  expressions  hold  for  M  and 
A''.     The  components  of  H'  are  therefore 

U  =  L-7iC  +  ^B,  M'  =  M  -  f  A  +  ^C,  N'  =^N-^B+7]A; 

and  its  direction  cosines  are 

H"    ^  ~  ^"    "  7  //'  ■ 

The  central  axis  being  defined  (Art.  223)  as  that  line  for 

which  the  vector  of  the  resulting  couple  is  parallel  to  the 

direction  of  the  resultant,  the  point  0'{^,  t],  ^)  will  lie  on  the 

central  axis  if  the  direction  cosines  of  H'  are  equal  to  those 

of  R,  viz.  to  a  =  A/R,  13  =^  B/R,  y  =  C/R.  Hence  the 
equations  of  the  central  axis  are 

L'  ^  ilf '  ^  N^ 
A  ~   B        C  ' 

that  is, 

L-r]C  -\-^B  ^  M  -  rA  +  ^C  _  N  -  ^B  -{-yA 
A  B  C 

6.  Constraints;  friction. 

231.  It  has  been  shown  in  Art.  229  that  the  number  of  the 
conditions  of  equilibrium  is  six,  for  a  rigid  body  that  is 
perfectly  free.  This  number  will  be  diminished  whenever 
the  body  is  subject  to  conditions  restricting  its  possible 
motions.  Such  conditions,  or  constraints,  may  be  of  various 
kinds;  the  body  may  have  a  fixed  point,  or  a  fixed  axis,  or 
one  of  its  points  may  be  constrained  to  move  along  a  given 
curve  or  to  remain  on  a  given  surface,  etc. 


232.]  STATICS  OF  THE  RIGID   BODY  179 

Now  a  free  -point  is  said  to  have  three  degrees  of  freedom 
because  its  position  is  determined  by  three  co-ordinates.  One 
conditional  equation  between  its  co-ordinates  restricts  the 
point  to  the  surface  represented  by  that  equation;  it  has 
then  one  constraint  and  two  degrees  of  freedom.  Two  con- 
ditions restrict  the  point  to  a  curve,  viz.  the  intersection  of 
the  two  surfaces  represented  by  the  two  equations  of  con- 
dition; the  point  then  has  two  constraints  and  one  degree  of 
freedom. 

The  position  of  a  rigid  body  is  determined  by  the  position 
of  any  three  of  its  points,  not  in  a  Hne,  i.  e.  by  nine  co-ordi- 
nates between  which,  however,  there  exist  three  conditions, 
expressing  the  constancy  of  the  distances  of  the  three  points. 
A  free  rigid  body  has  therefore  six  degrees  of  freedom,  since  six 
independent  quantities  determine  its  position. 

The  most  general  instantaneous  state  of  motion  that  a  free 
rigid  body  can  have  is  a  twist,  or  screw-motion  (Art  123), 
consisting  of  an  angular  velocity  about  a  certain  axis  and  a 
linear  velocity  along  this  axis;  each  of  these  velocities  has 
three  components  along  the  rectangular  axes,  and  these  six 
components  can  be  regarded  as  the  six  independent  possible 
motions  of  the  body,  on  account  of  which  it  is  said  to  have 
six  degrees  of  freedom. 

Equilibrium  will  exist  only  when  these  six  possible  motions 
are  prevented;  hence  there  must  be  six  conditions  of  equi- 
librium. 

232.  We  proceed  to  consider  some  forms  of  constraint 
and  the  corresponding  changes  in  the  equations  of  equi- 
Hbrium. 

It  is  often  convenient  in  dynamics  to  replace  such  re- 
straining conditions  by  forces,  usually  called  reactions. 
Whenever  it  is  possible  to  introduce  such  forces  having  the 


180 


STATICS 


[233. 


same  effect  as  the  given  conditions,  the  body  may  be  re- 
garded as  free,  and  the  general  equations  of  equihbrium 
can  be  apphed. 

233.  Rigid  Body  with  a  Fixed  Point.  A  body  that  is  free  to  turn 
about  a  fixed  point  A  can  be  regarded  as  free  if  tlie  reaction  A  of  this 
point  be  introduced  and  combined  with  the  other  forces  acting  on  the 
body. 

Let  Ax,  Ay,  Azhe  the  components  of  A;  then,  taking  the  fixed  point 
A  as  origin,  the  six  equations  of  equihbrium  (Art.  229)  are 


ZX  +Ax  =  0, 

27 +  Ay  =0, 

^Z  +A,  =0, 

^{yZ  -  zY)  =  0, 

S(2Z  -  xZ)  =  0, 

Z{xY  -  yX)  =  0. 

B, 


>fA, 


The  first  three  of  these  equations  serve  to  determine  the  reaction 

of  the  fixed  point;  tlie  last  three  are  the  actual  conditions  of  equilibrium 

corresponding  to  the  three  degrees  of  freedom  of  a  body  with  a  fixed 

point. 

Hence,  a  rigid  body  having  a  fixed  point  is  in  equilibrium  if  the  sum 

of  the  moments  of  all  the  forces  vanishes  for  any  three  non-com-planar  axes 

passing  through  the  fixed  point. 

234.  Rigid  Body  with  a  Fixed  Axis.     A  body  with  a  fixed  axis  has 

but  one  degree  of  freedom; 
indeed,  the  only  possible  mo- 
tion consists  in  rotation  about 
this  axis. 

An  axis  is  fixed  as  soon  as 
two  of  its  points,  say  A,  B, 
are  fixed.  Hence,  after  intro- 
ducing the  reactions  Ax,  Ay, 
Az,  Bx,  By,  Bz,  of  these  points, 
the  body  can  be  regarded  as 

free.     If  the  point  B  be  taken  as  origin,  the  line  BA  as  axis  of  z  (Fig. 

61),  the  equations  of  equilibrium  become 

SZ  +  Ax  +  fix  =  0,         27  +  A,i  +  By  =  0,         2Z  +  Az  +  Bz  =  0, 

-LiyZ  -  zY)  -  Aya  =  0,       2(2X  -  xZ)  -f  Am  =  0, 

2  (a: 7  -  yX)  =  0, 
where  a  =  BA. 

The  last  of  the  six  equations  is  the  only  independent  condition  of 


A.- 


y/B. 


Fig.  61. 


236.]  STATICS   OF   THE   RIGID   BODY  181 

equilibrium  of  the  constrained  body;  the  first  five  determine  Ax,  Bx, 
Ay,  By,  Az  +  Bz.  The  two  2-components  cannot  be  found  separately, 
since  they  act  in  the  same  straight  line. 

Hence,  a  rigid  body  having  a  fixed  axis  is  in  equilibrium  if  the  sum 
of  the  moments  of  all  the  forces  vanishes  for  the  fixed  axis. 

235.  If,  in  the  preceding  article,  the  axis  be  not  absolutely  fixed, 
but  only  fixed  in  direction  so  that  the  body  can  rotate  about  the  axis 
and  also  slide  along  it,  we  have  evidently 

Az  =  0,     Bz  =  0; 

hence,  by  the  third  equation  of  equilibrium. 

2Z  =  0, 

as  an  additional  condition  of  equilibrium. 

The  body  has  in  this  case  two  degrees  of  freedom. 

236.  Rigid  Body  with  a  Fixed  Plane.  A  body  constrained  to  slide 
on  a  fixed  plane  (that  is,  to  move  so  that  the  paths  of  all  its  points 
lie  in  parallel  planes)  has  three  degrees  of  freedom.  At  every  point  of 
contact  between  the  body  and  the  plane,  the  latter  exerts  a  reaction.  As 
all  these  reactions  are  parallel,  they  are  equivalent  to  a  single  resultant 
N.  Taking  the  fixed  plane  as  the  plane  xy,  N  will  be  parallel  to  the 
axis  of  z;  hence,  if  o,  b,  0  be  the  co-ordinates  of  its  point  of  application, 
the  six  equations  of  equilibrium  are 

2X  =  0,       Si'  =  0,       XZ  +  N  =  0, 

•ZiyZ  -  zY)  +bN  =  0,  i:(zX  -  xZ)  -  aN  =  0, 

Z{xY  -  ijX)  =  0. 

The  third,  fourth,  and  fifth  equations  determine  the  reaction  N 
and  the  co-ordinates  a,  b  of  its  point  of  apjjlication.  The  three  other 
equations  are  the  actual  conditions  of  equilibrium;  they  agree,  of  course, 
with  the  three  conditions  of  equilibrium  of  a  plane  system  as  found  in 
Art.  219. 

If  there  be  not  more  than  three  points  of  contact  (or  supports) 
between  the  body  and  the  fixed  plane,  the  reactions  of  these  points 
can  be  found  separately.  Let  Ai,  A2,  A3  be  the  three  points  of  contact; 
Ni,  N2,  Ns  the  required  reactions;  ai,  b^,  «•>,  b^,  az,  bs  the  co-ordinates 
of  Ai,  Ai,  As;  then  A^"  must  be  resolved  into  three  parallel  forces  passing 
through  these  points,  and  the  conditions  are 


182  STATICS  [237. 

Ni  +  N2  +  N3  =  N, 
UiNi  +  a^Ni  +  a3A'"3  =  aN , 
biNi  +  b2N2  +  hsNs  =  bN. 

These  three  equations  always  determine  A'^i,  N^,  N3.     For  if  the 
determinant  of  the  coefficients  of  Ni,  N2,  Ns  vanished, 


1        1       1 

cti     a2     as 

bi      bi      bs 


=  0, 


the  three  points  Ai,  A2,  A3  would  lie  in  a  straight  line,  and  hence  the 
body  would  not  be  properly  constrained. 

The  reactions  become  indeterminate  whenever  there  are  more  than 
three  points  of  contact. 

237.  Friction.  The  reaction  between  two  surfaces  in 
contact  has  so  far  been  regarded  as  directed  along  the -com- 
mon normal  of  the  surfaces  (Art.  195).  If  this  is  true  the 
surfaces  are  said  to  be  perfectly  smooth. 

The  surfaces  of  physical  bodies  are  rough,  i.  e.  they  pre- 
sent small  elevations  and  depressions;  when  two  such  sur- 
faces are  "  in  contact  "  the  projections  of  one  will  more 
or  less  enter  into  depressions  of  the  other;  the  greater  the 
normal  pressure  between  the  surfaces,  the  more  will  this 
be  the  case.  Hence  when  a  tangential  force  acting  on  one 
of  the  bodies  tends  to  slide  its  surface  over  that  of  the  other 
body,  a  resistance  will  be  developed  whose  magnitude 
must  depend  on  the  roughness  of  the  surfaces  and  on  the 
normal  pressure  between  them.  This  resistance  is  called 
the  force  of  sliding  friction,  or  simply  the  friction. 

The  study  of  friction  belongs  properly  to  applied  mechan- 
ics, and  will  here  only  be  touched  upon  very  briefly. 

238.  Imagine  a  body  resting  with  a  plane  surface  on  a 
horizontal  plane.  Let  a  small  horizontal  force  P  be  applied 
at  its  centroid  (which  is  supposed  to  be  situated  so  low  that 


239.]  STATICS   OF  THE   RIGID   BODY  183 

the  body  is  not  overturned),  and  let  the  force  P  be  gradu- 
ally increased  until  motion  ensues.  At  any  instant  before 
motion  sets  in,  the  friction  is  equal  to  the  value  of  P  at 
that  instant.  The  value  of  P  at  the  moment  when  motion 
just  begins  is  equal  and  opposite  to  the  frictional  resistance 
F  between  the  surfaces  at  this  moment,  and  this  resist- 
ance is  called  the  limiting  static  friction. 

Careful  experiments  with  dry  solids  in  contact  have 
shown  this  force  to  be  subject  to  the  following  laws: 

(1)  The  magnitude  of  the  limiting  friction  F  hears  a  con- 
stant ratio  to  the  normal  pressure  N  between  the  surfaces  in 
contact;  that  is 

F  =  fj^N, 

where  /i  is  a  constant  depending  on  the  condition  and  nature 
of  the  surfaces  in  contact.  This  constant  which  must  be 
determined  experimentally  for  different  substances  and 
surface  conditions  is  called  the  coeflEicient  of  static  friction. 
It  is  in  general  a  proper  fraction;  for  jDerfectly  smooth  sur- 
faces ;u  =  0. 

(2)  For  a  given  normal  pressure  the  limiting  static  friction, 
and  hence  the  coefficient  of  static  friction,  is  independent  of 
the  area  of  contact,  provided  the  pressure  be  not  so  great  as  to 
produce  cutting  or  crushing. 

239.  The  frictional  resistance  between  two  surfaces  in 
relative  motion  is  called  kinetic  friction.  It  is  subject,  in 
addition  to  the  two  laws  just  mentioned,  to  the  third  law: 

(3)  For  moderate  velocities,  kinetic  friction  is  nearly  inde- 
pendent of  the  velocities  of  the  bodies  in  contact. 

The  coefficient  of  static  friction  is  somewhat  greater  than 
that  of  kinetic  friction.  A  slight  jarring  will  often  reduce 
the  coefficient  from  its  static  to  its  kinetic  value. 


184 


STATICS 


1240. 


It  must  not  be  forgotten  that  these  so-called  laws  of 
friction  are  experimental  laws,  and  therefore  true  only  ap- 
proximately and  within  the  limits  of  the  experiments  from 
which  they  were  deduced.  When  the  relative  velocity  of 
the  surfaces  in  contact  is  high,  or  when,  as  is  usually  the 
case  in  machinery,  a  lubricating  material  is  introduced 
between  the  two  surfaces,  the  frictional  resistance  is  found 
to  depend  on  a  number  of  other  circumstances,  such  as  the 
temperature,  the  form  of  the  surfaces,  the  velocity,  the 
nature  of  the  lubricator,  etc.  Indeed,  when  the  supply  of 
the  lubricant  is  sufficient,  the  two  solid  surfaces  are  kept 
by  it  out  of  actual  contact;  the  coefficient  of  friction  in 
this  case  varies  w4th  the  pressure,  area  of  contact,  velocity, 
and  temperature. 

240.  Consider  again  a  body  resting  on  a  horizontal  plane  (Fig.  62) 
and  acted  upon  by  a  horizontal  force  P  just  large  enough  to  equal  the 

limiting  friction  F.  The  normal 
reaction  N  of  the  plane  is  equal  and 
opposite  to  the  weight  W.  The 
body  is  thus  in  equilibrium  under 
the  action  of  the  two  pairs  of  equal 
and  opposite  forces;  but  motion  will 
ensue  as  soon  as  P  is  increased.  If 
P  be  decreased,  F  will  decrease  at 
the  same  rate,  so  that  the  equilib- 
rium remains  undisturbed. 
The  force  of  friction  F  can  be  combined  with  the  normal  reaction 
A''  to  form  a  resultant, 


R 


VF^  +  m  =  VP^  +  TF2, 


which  represents  the  total  reaction  of  the  horizontal  plane. 

If  4,  be  the  angle  between  N  and  R  when  F  has  its  limiting  value 
F  =  fxN  (Art.  238),  we  have,  since  tan^  =  F/N, 


tan0  =  M- 


242. 


STATICS  OF   THE   RIGID   BODY 


185 


The  angle  ^  thus  furnishes  a  graphical  representation  for  the  coefficient 
of  friction  /x;  it  is  called  the  angle  of  friction. 

241.  If  the  plane  be  not  horizontal,  but  incHned  to  the  horizon  at 
an  angle  d,  the  weight  W  of  the  body  (regarded  as  a  particle)  resting 
on  the  plane  can  be  resolved  into 
a   component  W  sin9  along   the 
plane,  and  a  component  W  cos9 
perpendicular    to    it    (Fig    63). 
Hence,  if  no  other  forces  act  on 
the  body  it  will  be  in  equilibrium, 
provided  the  component  W  sin9 
be  not  greater  than  the  hmiting 
friction  F  =  nW  cos9.    The  limit- 
ing   condition    of   equilibrium   is 
therefore. 


ixW  cosO  =  W  sin0,     or      n  =  tan9; 


Fig.  63. 


in  other  words,  if  the  angle  0  be  gradually  increased,  the  body  will  not 
sUde  down  the  plane  until  d  >  <l>.  This  furnishes  an  experimental 
method  of  determining  the  angle  of  friction  <j),  which  on  this  account 
is  sometimes  called  the  angle  of  repose. 

242.  A  particle  P  (Fig.  64)  will  be  in  equilibrium  on  any  rough 
surface,  if  the  total  reaction  of  the  surface,  i.  e.  the  resultant  R  of  the 

normal  reaction  A'^  and  the 
friction  F,  is  equal  and  op- 
posite to  the  resultant  R'  of 
all  the  other  forces  acting  on 
the  particle. 

The   limiting  value  of  the 
angle  between  N  and  R  is  <j) 
so  that  the  particle  can  be  in 
equilibrium  only  if  the  result- 
ant R'  makes  with  the  normal 
an  angle  <0.   Hence,  if  about 
the  normal  FN  as  axis,  and 
with  P  as  vertex,  a  cone  be  described  whose  vertical  angle  is  2(^,  the  con- 
dition of  equilibrium  is  that  R'  must  lie  within  this  cone.     The  cone  is 
called  the  cone  of  friction. 


Fig.  64. 


186  STATICS  [243. 

243.  Exercises. 

(1)  A  weight  W  is  to  be  hauled  along  a  horizontal  plane,  the  coeffi- 
cient of  friction  being  fi  =  tan  </>.  Determine  the  required  tractive  force 
P  if  it  is  to  act  at  an  inclination  a  to  the  horizon,  and  show  that  this 
force  is  least  when  a.  =  4>. 

(2)  A  particle  ot  weight  W  is  in  equilibrium  on  a  rough  plane  in- 
clined to  the  horizon  at  an  angle  d,  under  the  action  of  a  force  /-*  parallel 
to  the  plane  along  its  greatest  slope.  Determine  P:  (a)  when  0  >  0, 
(6)  when  B  =  4>,  (c)  when  0  <  (j>,  <j>  =  tan'^/u  being  the  angle  of  friction. 

(3)  Solve  Ex.  (2)  (o)  graphically  by  means  of  the  friction  angle 
and  determine  what  part  of  P  is  required  to  overcome  friction. 

(4)  A  body  weighing  240  pounds  is  pulled  up  a  plane  inclined  at 
45°,  by  means  of  a  rope.  If  m  =  li,  find  the  tension  of  the  rope. 
What  portion  of  it  is  due  to  friction? 

(5)  A  homogeneous  straight  rod  AB  =  21  of  weight  W  rests  with 
one  end  A  on  a  horizontal  floor,  with  the  other  end  B  against  a  vertical 
wall  whose  plane  is  at  right  angles  to  the  vertical  plane  of  the  rod. 
If  there  be  friction  of  angle  4>  at  both  ends,  determine  the  limiting 
position  of  equilibrium. 

(6)  A  straight  homogeneous  rod  AB  =21,  of  weight  W,  rests  with 
the  lower  end  A  on  a  rough  horizontal  plane  and  with  the  point  C 
{AC  =  c)  on  a  smooth  cylindrical  support.  The  rod  is  in  equilibrium 
when  inclined  at  a  given  angle  6  to  the  horizon ;  determine  the  coefficient 
of  friction  at  A  and  the  reactions  at  A  and  C. 

(7)  If  in  Ex.  (6)  there  be  friction  both  at  A  and  C,  the  friction  angle 
<^  being  the  same,  find  the  position  of  equilibrium  and  the  reactions 
at  A  and  C. 

(8)  A  solid  homogeneous  hemisphere  is  placed  with  its  curved  surface 
on  a  rough  inclined  plane;  investigate  the  conditions  of  equilibrium. 


CHAPTER  XII. 
THEORY  OF  ATTRACTIVE  FORCES. 

1.  Attraction. 

244.  Among  the  various  kinds  of  forces  introduced  in 
physics  for  describing  and  interpreting  natural  phenomena, 
forces  of  attraction  and  repulsion  occupy  a  most  prominent 
place. 

According  to  Newton's  law  (the  law  of  universal  or 
cosmical  gravitation,  the  "  law  of  nature  ")  every  particle 
of  matter  attracts  every  other  such  particle  with  a  force 
proportional  to  the  masses  and  inversely  proportional  to 
the  square  of  the  distance  of  the  particles,  and  this  force 
acts  along  the  line  joining  the  particles. 

Thus,  if  m,  m'  are  the  masses  of  the  particles,  r  their  dis- 
tance, and  K  a  constant,  the   force  with  which  m  attracts 

m'  and  fn'  attracts  in  is 

^         7nm' 

F    =    K-    2     . 

Each  particle  is  here  regarded  as  a  mass  concentrated  at  a 
point;  otherwise  we  could  not  speak  of  the  distance  of  the 
particles  and  of  the  line  joining  them  (comp.  Art.  156). 
As  the  distance  r  approaches  zero,  the  magnitude  of  the 
force  F  becomes  infinite  and  its  direction  indeterminate. 

245.  In  the  theory  of  gravitation,  the  masses  m,  m'  are 
essentially  positive.  The  constant  k,  called  the  constant 
of  gravitation,  evidently  represents  the  force  with  which 
two  particles,  each  of  mass  1 ,  attract  each  other  when  at  the 

187 


188  STATICS  [246. 

distance  1.  It  is  a  physical  constant  to  be  determined  by 
experiment,  and  its  numerical  value  depends  on  the  units 
of  measurement  adopted  for  mass,  length,  and  time. 

What  can  be  directly  observed  is  of  course  not  the  force 
itself,  but  the  acceleration  it  produces.  Dividing  the  force 
F  (Art.  244)  by  the  mass  m  of  the  attracted  particle  on 
which  it  acts  we  have  the  acceleration  j  produced  by  the 
force  with  which  m'  attracts  m  at  the  distance  r  from  rn: 

m' 

246.  It  will  be  shown  later  (Art.  253)  that  the  attraction  of  a  homo- 
geneous sphere  at  any  external  point  is  the  same  as  if  the  mass  of  the 
sphere  were  concentrated  at  its  center.  Hence  if  m'  be  the  mass  of 
the  earth  (here  regarded  as  a  homogeneous  sphere)  the  acceleration  it 
produces  in  any  mass  m  concentrated  at  a  point  P  above  its  surface, 
at  the  distance  OP  =  r  from  the  center  0,  is  j  =  Km'/r^.  Now  for 
points  near  the  earth's  surface  this  acceleration  is  known  from  experi- 
ments; it  is  the  acceleration  ^  of  a  body  falling  in  vacuo  (apart  from 
the  slight  effect  due  to  the  earth's  rotation,  see  Arts.  334,  461).  Hence, 
taking  the  radius  of  the  earth  as  r  =  6.37  X  10«  cm.,  its  mean  density 
as  p  =  5K,  and  g  =  980  cm./sec.^,  we  find  in  C.G.S.  units 

K  =  6.7  X  10-8. 

This,  then,  is  the  force  in  dynes  with  which  two  masses,  of  1  gram  each, 
would  attract  each  other  if  concentrated  at  two  points  1  cm.  apart. 
Conversely,  the  mean  density  of  the  earth  can  be  found  with  con- 
siderable accuracy  by  a  direct  experimental  determination  of  the  attrac- 
tion of  gravitation  between  two  given  masses  at  a  given  distance. 

247.  Exercises. 

(1)  With  r  =  3960  miles,  g  =  32  ft. /sec. 2,  p  =  5H,  show  that  the 
attraction  between  two  masses  of  1  lb.  each,  at  a  distance  of  1  ft.,  is 
equal  to  the  weight  of  0.33  X  lO-i"  lb. 

(2)  In  astronomy  and  in  the  general  theory  of  attraction  it  is  con- 
venient to  take  the  unit  of  mass  so  that  k  =  1.  Show  that  this  astro- 
nomical unit  of  mass,  i.  e.  the  mass  which  acting  on  an  equal  mass  at 
unit  distance  would  produce  unit  acceleration,  is  =  l//c. 


248.]  THEORY  OF   ATTRACTIVE  FORCES  189 

(3)  Show  that  k  =  1  if,  with  the  ordinary  unit  of  mass,  the  unit  of 
time  be  taken  as  3862  sec.     This  has  been  called  the  "natural  hour." 

248.  If  more  than  two  particles  are  given  the  forces  of 
attraction  exerted  on  any  one  of  the  particles,  m,  being  con- 
current are  equivalent  to  a  single  resultant.  This  resultant, 
divided  by  the  mass  m  of  the  attracted  particle,  is  called  the 
attraction  at  the  point  P  where  m  is  situated. 

If,  instead  of  a  finite  number  of  particles,  any  continuously 
distributed  masses  of  one,  two,  or  three  dimensions  (Art. 
155)  are  given  they  can  be  resolved  into  elements  which  in 
the  limit  can  be  regarded  as  particles.  The  first  problem  in 
the  theory  of  attraction  consists  in  determining  the  attraction 
at  any  point,  due  to  any  given  masses. 

Notice  that  the  ''attraction  at  any  point,"  as  thus  defined, 
has  the  dimensions  of  an  acceleration  and  not  of  a  force. 

Let  P(x,  y,  z)  be  the  attracted  point  of  mass  1,  dm'  an 
element  of  the  attracting  masses  at  Q{x' ,  y' ,  z'),  PQ  =  r 
the  distance  of  these  points;  then  the  attraction  at  P  due 
to  dm'  is  Kdm'/r^,  and  if  a,  |3,  y  are  its  direction  cosines,  its 
components  are  Kadm'/r^,  K^dm'/r^,  Kydm'/r^.  Hence  the 
attraction  A  at  P,  due  to  all  the  given  masses,  has  the  rec- 
tangular components: 


X 


/adm'       ,,  rfidm'        „  rydm' 


with  r^  =  {x'  —  xy  -\-  {y'  —  y)~  +  {z'  —  z)-,  the  integrations 
extending  over  all  the  masses.  The  attraction  A  itself  and 
its  direction  cosines  I,  m,  n  are: 

X  Y  Z 


A^  ^X'-  +  Y^  +  Z\     I  =~,     m==-r,     n=  ,  . 
'  A  '  A  '  A 

It  is  in  general  most  convenient  to  take  the  attracted  point 
P  as  origin  so  that  r"^  =  x'"^  +  y'-  +  z'- 


190  STATICS  [249. 

249.  If  the  point  P  were  situated  within  the  attracting 
masses,  l/r^  would  become  infinite  within  the  hmits  of  integra- 
tion; hence  a  special  investigation  would  be  necessary  to 
determine  whether  the  integrals  representing  A",  Y,  Z  have 
a  meaning.  It  can  be  shown  without  difficulty  in  the  case 
of  three-dimensional  masses  that  the  integrals  have  a  meaning 
and  represent  the  attraction  even  at  an  internal  point  P. 
But  for  the  sake  of  simplicity,  we  here  confine  ourselves  to 
the  external  field.  In  other  words,  we  assume,  when  nothing 
is  said  to  the  contrary,  that  P  is  an  external  'point,  i.  e.  a 
point  such  that  a  sphere  can  be  described  about  it  such  as  not 
to  contain  within  it  any  portion  of  the  attracting  matter 
(except  the  unit  mass  at  P  itself). 

250.  The  problem  of  attraction  can  be  generalized  in 
various  ways.  Thus,  in  electricity  and  magnetism,  we  have 
to  consider  both  positive  and  negative  masses,  and  the  force 
may  be  a  repulsion  as  well  as  an  attraction.  The  force  be- 
tween two  electric  charges  as  well  as  that  between  two 
magnetic  poles  follows  Newton's  law  (Art.  244);  i.  e.  the 
force  is  directly  proportional  to  the  charges,  or  pole-strengths, 
and  inversely  proportional  to  the  square  of  the  distance. 
But  the  constant  k  has  a  very  different  value.  It  is  cus- 
tomary to  select  the  units  of  electric  charge  and  magnetic 
pole-strength  so  that  /c  =  1. 

It  is  sometimes  necessary  to  consider  forces  that  do  not  fol- 
low Newton's  law  of  the  distance.  Indeed,  Newtonian  attrac- 
tion is  merely  a  particular  case  of  the  more  general  type  of  force 

F  =  Kmm'fir), 

viz.  the  case  when/(r)  =  1/r^. 

251.  Spherical  shell.  The  attraction  due  to  a  mass  spread  uniformly 
over  a  sphere  is  zero  at  any  point  within  the  sphere,  while  at  any  outside 
point  it  is  the  same  as  if  the  mass  were  concentrated  at  the  center. 


251.] 


THEORY   OF   ATTRACTIVE   FORCES 


191 


Geometrical  method,  (a)  Attraction  at  an  inside  point.  Let  C  be 
the  center,  a  the  radius  of  the  sphere  (Fig.  65).  A  cone  of  vertex  P 
and  solid  angle  dw  {i.  e.  cutting 
out  an  area  element  dw  on  the 
sphere  of  radius  1  about  P  as  cen- 
ter) cuts  out  on  the  given  sphere 
a  surface  element  da  at  Q  and  a 
surface  element  da'  at  Q'.  It  wUl 
be  shown  that  the  mass  elements 
on  these  surface  elements  produce 
equal  and  opposite  attractions  at 
P.  As  the  whole  sphere  can  thus 
be  divided  into  pairs  of  elements 
whose  attractions  at  P  balance  it 
follows  that  the  attraction  at  P  is 
zero. 

Put  PQ  =  r,  PQ'  =  r'\  on  the  sphere  of  radius  r  about  P  the  cone 
cuts  out  an  element  r-c/co  at  Q,  and  we  have  evidently  da  =  r^dw/cosCQP; 
hence  if  the  surface  density  is  p',  the  mass  on  da  is  p'r'^dw/cosCQP,  and 
the  attraction  at  P  due  to  this  mass  is  up'dw/cosCQP.  In  the  same 
way  we  find  that  the  mass  on  da'  at  Q'  produces  at  P  the  attraction 


Fig.  66. 

Kp'du/cosCQ'P.     As  for  the  sphere  the  angles  CQP  and  CQ'P  are  equal, 
the  attractions  are  equal. 

(6)  Attraction  at  an  outside  point.     Let  P'  (Fig.  66)  be  the  point 


192  STATICS  [251. 

inverse  to  P  with  respect  to  the  given  sphere,  i.  e.  the  point  P'  on  CP 
such  that  if  CP  =  p,  CP'  =  p',  we  have 

pp'  =  a^. 

The  extremities  Q,  Q'  of  any  cliord  tlirough  P'  determine  with  C,  P,  P' 

two  pairs  of  similar  triangles:  CQP'  and  CPQ,  CQ'P'  and  CPQ';  for, 

each  pair  has  the  angle  at  C  in  common  and  the  sides  including  the 

equal  angles  proportional  owing  to  the  relations  pp'  =  a^,  CQ  =  CQ' 

=  a.    It  follows  that  2^  CQP'  =  CPQ,  2^  CQ'P'  =  CPQ';  hence,  as  the 

triangle  QCQ'  is  isosceles,  the  line  CP  bisects  the  angle  QPQ'. 

With  the  aid  of  these  geometrical  properties  it  can  be  shown  that 

equal  attractions  are  produced  at  P  by  the  masses  on  the  elements 

da  at  Q  and  da'  at  Q',  cut  out  by  a  cone  of  solid  angle  dw  with  vertex 

at  the  point  P'  inverse  to  P.     For  the  mass  elements  at  Q,  Q'  we  have 

as  in  the  case  (a) : 

,      r-dci  ,         ,  ,  ,         ,      r'-dw 

dm=pda=p  ^^^^Q^,     dm    =pdcr    =p  -^^^jq^,  , 

where  r  =  P'Q,  r'  =  P'Q'.  Hence  the  corresponding  attractions  at 
P  are: 

Kp'rHoi  Kp'r"^do} 

PQ^  cosCQP' '     PQ'2  cosCQ'P" 

and  these  are  equal,  since  for  the  sphere  ^  CQP'  =  CQ'P',  and  the 
similar  triangles  give 

r a         ^'    _  i? 

PQ~  p'     W  ~  P' 

As  shown  above;  these  attractions  make  equal  angles  with  PC',  hence 
their  components  along  this  line  are  equal  while  their  components  at 
right  angles  to  CP  are  equal  and  opposite.  The  two  elements  da  at  Q 
and  da'  at  Q'  produce  therefore  together  at  P  an  attraction  along  PC 
equal  to 

2/cp'rt^dw 

V' 

The  coefficient  of  dw  is  constant;  the  summation  over  the  unit  sphere 
gives  J"dw  =  2ir,  since  a  double  cone  was  used.  Hence  the  total 
attraction  at  P  is 

A  A  lO^  "*' 

P'  p^ 


252.] 


THEORY   OF   ATTRACTIVE   FORCES 


19.3 


where  m'  =  47rp'a-  is  the  whole  mass  on  the  sphere.  This  shows  that 
the  attraction  is  the  same  as  if  this  mass  were  concentrated  at  the  center 
of  the  sphere. 

(c)  AttracHon  at  a  poinl  on  the 
sphere.  If  the  point  P  approaches 
the  surface  from  within  the  attrac- 
tion remains  constantly  zero;  if 
P  approaches  the  surface  from 
without  the  attraction  approaches 
the  limit  Km'/a'^  =  iwKp'.  At  a 
point  P  on  the  sphere  (Fig.  67) 
the  attraction  can  be  shown  to  be 

A   =  27r/cp'. 

For,  the  mass  on  da  at  Q  is  p'da  =  Fig.  67 

p'r^du/cosCQP;  its  attraction  at  P 

is  =  Kp'dcj/cosCQP,  and  as  the  angles  at  P  and  Q  are  equal,  the  projection 
of  tliis  attraction  on  PC  is  Kp'doi.  As  P  lies  on  the  surface,  f  doi  =  2-k; 
hence  the  total  attraction  is  =  I-kkp  . 

The  attraction  exerted  by  the  whole  mass  on  the  mass  element  p'da 
situated  at  P  is  of  course  =  2i^Kp''^d<j. 

252.  Analytical  method.     Whether  P  lies  inside  or  outside  the  sphere 
we  take  P  as  origin,  PC  as  polar  axii?,  and  put  PQ  =  r,  ^  PCQ  =  </>, 


Fig.  6S. 

Q  being  any  point  of  the  sphere  (Fig.  68),  and  as  before  CP  =  p, 
CQ  =  a.  As  mass  element  take  the  mass  p'  ■  2x0  sin<^  •  ad(t>,  contained 
between  the  plane  through  Q  at  right  angles  to  PC  and  an  infinitely  near 
parallel  plane.  The  attraction  produced  at  P  by  this  element  is 
directed  along  PC  and 
14 


194  STATICS  1254. 

,  „   .      ,      cosCPQ      „      ,  „  .      J      p  —  a  coscf) 
=  2Tr Kp  a-  sm<l>a<l>  ■ ^ —  =  ^ttkt)  a-  sin<^a0  • ^ , 

where 

J.2  =  q2  _j_  p'i  _  2ap  co.s</), 

and  hence 

rdr  =  ap  &in0  d</). 

Substituting    for   asin^  r/0  and  ooos</>    their    expressions    from    these 
relations  we  find  for  the  attraction  of  the  ring  element  at  P: 

,  a  p^  —  a-  +  r'^  J 
TTKp'  — •  dr. 

(a)  For  an  inside  point  P  we  have  p  <  a,  and  the  limits  of  integration 
for  r  are  from  a  —  p  to  a  +  p.     Hence  the  resultant  attraction  at  P  is 

A  =  TTKp'     -   I         I  —    ,       +  1  I  f/r  =  TT/v'p'    ,  I  +r)        =0. 

p2  J„-p   \      r-  J  p'  \       r  J.,-p 

For  an  outside  point  P  we  have  p  >  a  and  the  limits  are  from  p  —  a 
to  p  +  a;  hence 

,    a  / a-  —  p-   ,     \?'+«        ,       ,  d?  vn' 

A     =    TTKp'         ,  —    +r)  =    iTTKp'   -   ,   =    K  -  Y  ■ 

p^  \        r  Jp-a  p^  p^ 

253.  From  the  results  of  Arts.  251,  252,  it  readily  follows  that  the 
attraction  due  to  a  homogeneous  solid  shell  (mass  between  twoconcentric 
spheres)  is  zero  within  the  hollow  of  the  shell,  while  at  an  outside  point 
it  is  the  same  as  if  the  mass  were  concentrated  at  the  center.  It  suffices 
to  resolve  the  shell  into  concentric  shells  of  infinitesimal  thickness  da 
and  put  p'da  =  p,  the  volume  density. 

In  particular,  for  a  homogeneous  solid  sphere  of  radius  a  and  volume 
density  p  the  attraction  at  the  distance  p  >  a  from  the  center  is 

.  ?n'       4        ,  o3 

p^        6  p^ 

254.  Exercises, 

(1)  Show  that  the  results  of  Art.  253  hold  for  a  solid  shell  whose 
density  is  any  function  of  the  distance  from  the  center. 

(2)  By  Art.  252,  the  attraction  due  to  a  mass  distributed  uniformly 
over  a  sphere  when  considered  as  a  fimction  of  p  has  a  point  of  dis- 
continuity; illustrate  this  by  a  sketch. 

(3)  Prove  that  the  attraction  at  the  center  due  to  a  mass  distributed 
uniformly  along  a  circular  arc  of  angle  la  and  radius  a  is  =  2/cp"  sina/a; 
show  that  a  mass  equal  to  that  of  the  chord,  if  it  had  the  same  density 


254.1  THEORY  OF  ATTRACTIVE   FORCES  195 

p",  placed  at  the  midpoint  of  the  arc,  would  produce  the  same  attraction 
at  the  center. 

(4)  Prove  that  the  attraction  of  a  homogeneous  rectilinear  segment 
A1A2,  at  a  point  P  whose  perpendicular  distance  PO  from  A1A2  makes 
the  angles  di,  62  with  PAi,  PA2,  bisects  the  angle  A1PA2  and  has  the 
value  2kp"  sins  (^2  —  6i)/p.  Show  that  the  arc  of  the  circle  of  radius 
PO  =  p  about  P,  bounded  by  PAi  and  PA2,  if  of  the  same  density, 
produces  at  P  the  same  attraction. 

(5)  Show  that  in  any  plane  through  A1A2  the  confocal  hyperbolas 
having  A1A2  as  foci  are  the  lines  of  force  in  the  field  of  the  rectilinear 
segment;  i.  e.  they  have  the  property  that  the  attraction  at  any  point 
P  is  tangent  to  the  hyperbola  through  P. 

(6)  Show  that  for  a  homogeneous  rod  of  infinite  length  the  attraction 
at  any  point  is  normal  to  the  rod  and  inversely  proportional  to  the 
distance  from  the  rod.  Hence  show  that  the  attraction  due  to  a 
homogeneous  circular  cylinder,  of  radius  a  and  infinite  length,  at  any 
point  P  at  the  distance  PC  =  p  >  a  from    the  axis,  is    =  2irKpa-/p. 

(7)  Prove  that  the  attraction  due  to  a  mass  spread  uniformly  over 
the  area  of  a  circle  of  radius  a,  at  a  point  P  on  the  axis  of  the  circle,  at 
the  distance  PC  =  p  from  the  center  C,  is  =  2irKp'il  —  p/Va^  +  P')- 

(8)  Two  parallel  homogeneous  straight  rods  of  equal  density  p"  are 
placed  so  that  the  line  joining  their  midpoints  is  at  right  angles  to  each; 
if  their  lengths  are  2a,  25,  and  their  distance  is  c,  find  their  mutual  attrac- 
tion, i.  e.  the  force  required  to  hold  them  apart. 

(9)  Show  that  the  attraction  exerted  by  a  homogeneous  right  circular 
cone  of  vertical  angle  2a  and  height  h,  at  the  vertex,  is  =  2irKph{l  — 
cosa).  Show  that  the  same  expression  holds  for  a  frustum  of  height  h 
and  angle  2a. 

(10)  Two  equal  circular  disks,  of  radius  a,  are  placed  at  right  angles 
to  the  line  joining  their  centers  whose  distance  is  c.  If  one  attracts 
while  the  other  repels,  determine  the  resultant  force  at  a  point  P  on 
the  line  of  the  centers,  at  a  distance  p  from  the  nearer  center.  Wliat 
becomes  of  this  force  when  c  is  indefinitely  diminished? 

(11)  Show  that  the  attraction  of  a  homogeneous  solid  hemisphere 
at  a  point  on  its  edge  is  =  f^Kpa  i/tt^  +  4,  and  that  it  is  inclined  to 
the  base  at  an  angle  of  about  32J^°. 


196 


STATICS 


[255. 


2.  The  potential. 

255.  As  shown  in  Art.  248,  the  determination  of  the  at- 
traction, due  to  given  masses,  at  any  particular  point  P  is  a 
mere  problem  of  integration.  The  next  problem  that  pre- 
sents itself  in  the  theory  of  attraction  is  to  express  the 
attraction  A  as  a  function  of  the  point  P,  or  rather  the  com- 
ponents X,  Y,Z  oi  A  as  functions  of  the  co-ordinates  x,  y,  z 
of  P,  and  to  study  the  nature  of  these  functions.  The  solu- 
tion of  this  problem  is  greatly  facilitated  by  observing  that 
there  exists  a  function  C/,  known  as  the  potential  of  the  given 
masses,  which  has  the  property  that  the  comyonents  of  A  are 
its  first  partial  derivatives : 

dU 


dU 

dy' 


Z  = 


dz 


A  function  having  this  property  may  exist  for  forces  that 
are  not  Newtonian  attractions;  it  is  then  called  a  force- 


function.     Forces  for  which  a  force-function  exists  are  called 
conservative  forces. 

256.  Let  us  consider  the  most  simple  case  of  Newtonian 
attraction,  viz.  the  field  generated  by  a  single  particle  m', 
situated  at  Q  (Fig.  69).     The  attraction  at  P(x,  y,  z),  due 


257.]  THEORY   OF  ATTRACTIVE   FORCES  197 

to  m'  at  Q,{x\  y',  z')  is  A  =  kvi' Ir"^,  where  r'^  =  {x  —  x'Y  + 
{y  —  y'Y  -{-  iz  —  z'y.  As  this  attraction  has  the  sense 
from  P  toward  Q,  its  direction  cosines  are  —  {x  —  x')lr, 
—  {y  —  y')l'>'j  —  (2  —  z')lr;  hence  the  rectangular  components 
of  the  attraction  are : 

X  =-  Kill'  ^^',     Y  =-  Km'  '^^,    Z  =-  km'  ^^. 

It  is  easily  verified  that  these  expressions  are  the  partial 
derivatives  with  respect  to  x,  y,  z  of  one  and  the  same  func- 
tion, viz. 

r 

this  then  is  the  potential  of  a  single  particle  m'. 

257.  Notice  that  this  function  is  one-valued  and  con- 
tinuous throughout  the  whole  of  space,  except  at  Q  where  it 
has  a  simple  pole  {i.  e.  U  becomes  infinite  like  1/r  for  r  =  0), 
and  that  it  vanishes  at  infinity.  The  same  properties  hold  for 
all  derivatives  of  U  except  that  Q  becomes  a  pole  of  higher 
order. 

For  the  projection  of  the  attraction  A  on  any  direction  s 
we  have 

.    _   ydx       ydy       ydz  __  dU dx      dU dy      dU dz  _  dU  _ 
ds  ds  ds      dx  ds      dy  ds      dz  ds      ds    ' 

i.  e.  the  s-component  of  A  is  the  s-dcrivative  of  U. 

For  the  second  x-derivative  of  U  we  have  since  dr/dx  = 
(x  —  x')/r: 

§^  _dX  _  _       ,  /  1  X  -  x'dr\ 

dx^        dx  \r^  r*      dx/ 

,ri    3(x  -  x'yi 

and  similarly: 


198  STATICS  [258. 


STATICS 

d'~U      dY 
dy'       dy 

—  Km' 

1 

3(?/  -  y'Y 

dz"-        dz 

—  Km' 

"1 

3(z-  z'y~\ 

^6 

■]■ 


Adding  and  observing  that  (x  —  x')~  +  (?/  —  ?yO^  +  (^  —  2')^ 
=  r'  we  find 

dnj    SHI    dHj  ^  0 

dx'        dy-        dz' 

This  equation,  satisfied  l^y  the  potential  at  every  point 
excepting  the  point  Q  where  the  attracting  mass  m'  is  situated, 
is  known  as  Laplace's  equation,  or  the  potential  equation. 

258.  These  results  are  readily  generalized.  If  the  field  is 
due  to  any  finite  number  of  particles  m/,  rih',  •  •  •  at  the 
distances  ri,  r^,  •  •  •  from  the  attracted  point  P,  their  potential 
is  defined  as 

^,         /CW?i'    ,     KfUo'    ,  ^  Km' 

*     U  = 1 ■  +  •  •  •  =  -S . 

ri  7-2  r 

If  the  field  is  due  to  continuous  masses  their  potential  is 

\bn' 


U 


rcbv/ 


For,  as  the  limits  of  integration  are  constant  the  derivatives 
of  U  with  respect  to  x,  y,  z  can  be  found  ])y  differentiating 
under  the  integral  sign;  we  have  therefore,  at  any  rate  at 
any  external  point  P\ 

dx  J      r^  dy  J       r^ 

du       r^-^'j  . 

-dz=-\l   ^^^^^' 

where  the  right-hand  members  are  evidently  the  components 
X,  Y,  Z  of  the  attraction  at  P. 


259.]  THEORY   OF  ATTRACTIVE   FORCES  199 

For  masses  of  finite  density  and  not  extending  to  infinity 
it  is  not  diflacult  to  show  that  the  function  U  has  a  single 
definite  finite  value  at  every  point  P  external  (and  even 
internal)  to  the  given  masses  and  that  it  is  a  continuous 
function  of  x,  y,  z. 

As  in  Art.  257  it  can  be  shown  that,  at  any  external  point, 
U  satisfies  Laplace's  equation 

d^U   ,   d^U      dHJ 
dx^         dy-         dz~ 

259.  The  potential  is  a  scalar  point-function;  i.  e.  it  is  not 
a  vector,  but  its  value  at  any  point  is  given  by  a  single  real 
number. 

The  locus  of  those  points  at  which  the  potential  U  has  a 
constant  value  c,  i.  e.  the  surface 

U  =  c, 

is  called  an  equipotential  surface  (level,  or  equilibrium,  sur- 
face) . 

As  the  first  derivatives  of  U  with  respect  to  x,  y,  z  are  on 
the  one  hand  equal  to  the  components  of  the  attraction 
while,  on  the  other,  they  are  proportional  to  the  direction 
cosines  of  the  normal  to  the  surface  U  =  c,  it  follows  that 
the  attraction  A  at  any  external  point  P  is  normal  to  the  equi- 
potential surface  passing  through  P. 

In  the  language  of  vector  analysis,  the  attraction  A  is  the 
gradient  of  the  potential  U. 

The  orthogonal  trajectories  of  the  family  of  equipotential 
surfaces  U  =  c  are  called  lines  of  force  since  each  of  these 
curves  has  the  property  that  the  tangent  at  any  one  of  its 
points  has  the  direction  of  the  attraction  at  that  point.  The 
differential  equations  of  the  lines  of  force  are  evidently 


200  STATICS  [260. 

1  lid  dx  _  dy  _  dz 

dx         dy         dz 
260.  Exercises. 

(1)  For  a  mass  spread  uniformly  over  the  surface  of  a  sphere  prove 
that,  witliin  the  sphere,  the  potential  is  zero  while,  outside  the  sphere, 
it  is  the  same  as  if  the  mass  were  concentrated  at  the  center.  Hence 
deduce  the  corresponding  results  for  a  homogeneous  solid  spherical 
shell. 

(2)  A  mass  is  distributed  uniformly  along  the  arc  of  a  parabola 
bounded  by  the  latus  rectum  4a;  show  that  at  the  focus  the  potential 
is  =  3.5245  Kp"  and  the  attraction  is  =  1.8856  Kp"la. 

(3)  Find  the  potential  due  to  a  homogeneous  circular  plate,  of 
radius  a,  at  a  point  P  of  its  axis,  at  the  distance  x  from  the  plate. 

(4)  Determine  the  equipotential  surfaces  for  a  straight  homogeneous 
rod;  comp.  Art.  254,  Ex.  4  and  5. 

(5)  For  a  mass  distributed  uniformly  along  the  circumference,  of  a 
circle,  determine  the  potential  at  any  point  in  the  plane  of  the  circle, 
and  show  that  at  a  distance  from  the  center  equal  to  %  the  radius  it  is 
=  7.2418  Kp". 

(6)  Show  that  a  force-function  exists  when  the  resultant  force  is  con- 
stant in  magnitude  and  direction. 

(7)  Find  the  force-function  in  the  case  of  a  free  particle  moving 
under  the  action  of  the  constant  force  of  gravity  (projectile  in  vacuo); 
determine  the  equipotential  surfaces. 

(8)  Show  the  existence  of  a  force-function  when  the  direction  of  the 
resultant  force  is  constantly  perpendicular  to  a  fixed  plane,  say  the 
a;y-plane,  and  its  magnitude  is  a  given  function  /(z)  of  the  distance  z 
from  the  plane. 

(9)  Find  the  force-function,  the  equipotential  surfaces,  and  the 
kinetic  energy  when  the  force  is  a  function  /(r)  of  the  perpendicular 
distance  r  from  a  fixed  line,  and  is  directed  towards  this  line  at  right 
angles  to  it. 

■  (10)  Show  the  existence  of  a  force-function  for  a  central  force,  i.  e.  a 
force  passing  through  a  fixed  point  (.xo,  yo,  zo),  if  the  force  is  a  function 
of  the  distance  r  from  this  point.     What  are  the  level  surfaces? 

(11)  Show  that  a  force-function  exists  when  a  particle  moves  under 
the  action  of  any  number  of  such  central  forces  as  in  Ex.  (10). 


262.1  THEORY   OF  ATTRACTIVE   FORCES  201 

3.  Virtual  work. 

261.  The  importance  of  the  potential  in  the  theory  of 
attraction  and  of  the  force-function  for  any  conservative 
forces  (Art.  255)  is  largely  due  to  their  connection  with  the 
idea  of  work. 

The  work  W  of  a  constant  force  F  in  a  rectilinear  displaces^ 
ment  s  of  its  point  of  application  is  defined  as  the  product 
of  the  projection  oi  F  on  s  into  s: 

W  =  Fs  cosi/',  "'""^ 

where  \f/  is  the  angle  l^etween  the  vectors  F  and  s.  In  other 
words,  work  is  the  dot-product  (Art.  141)  of  force  and  dis- 
placement : 

W  =  F-s. 

Thus,  e.  g.  when  a  body  of  weight  F  =  mg  slides  down  the 
greatest  slope  of  a  smooth  plane  inclined  at  the  angle  6  to 
the  horizon,  through  a  distance  s,  the  work  of  the  vertical 

force  F  is 

Fs  cos(^7r  -  e)  =  Fs  sin0  =  Fh, 

where  h  =  s  smd  is  the  vertical  height  through  which  the 
body  has  descended. 

It  follows  from  the  theory  of  projection  (Art.  198)  that 
the  work  of  a  force  is  the  sum  of  the  works  of  its  components. 
Hence,  if  X,  Y,  Z  are  the  rectangular  components  of  F, 
X,  y,  z  those  of  s,  we  have  (comp.  Art.  141) 

W  =  Xx-^  Yxj  +  Zz. 

262.  Work  is  not  a  vector,  but  a  scalar  quantity  (Art. 
259).  If,  in  the  definition  of  Art.  261,  we  take  for  ^p  the  lesser 
of  the  two  angles  made  by  the  vectors  F  and  s,  the  work  is 
positive  or  negative  according  as  i/'  is  <  or  >  ^r 

The  dimensions  of  work  are  evidently  ML'^T~^. 


202  STATICS  [263. 

The  unit  of  work  is  the  work  of  a  unit  force  (poundal, 
dyne)  through  a  unit  distance  (foot,  centimeter).  Tlie  unit 
of  work  in  the  F.P.S.  system  is  called  the  f oot-poundal ;  in 
the  C.G.S.  system,  the  erg.  Thus,  the  erg  is  the  amount  of 
work  done  by  a  force  of  one  dyne  acting  through  a  distance 
of  one  centimeter.     These  are  the  scientific  units. 

In  the  gravitation  system  where  the  pound,  or  the  kilogram 
is  taken  as  unit  of  force,  the  British  unit  of  work  is  the  foot- 
pound, while  in  the  metric  system  it  is  customary  to  use  the 
kilogram-meter  as  unit. 

263.  The  numerical  relations  between  these  units  are  obtained  as 

follows.     Let  X  be  the  number  of  ergs  in  the  foot-poundal,  then  (comp. 

Art.  175), 

em.  cm.2      ^    lb.  ft.^ 

sec.^  sec.2 

hence 

lb      /  ft   V 
a;  =  i^  .  (  '^-   )  =  4.2141  X  10^; 
gm.    Vcm./ 

i.  e.  1  foot-poundal  =  4.2141  X  10^  ergs,  and  1  erg  =  2.3730  X  IQ-^ 
foot-poundals. 

Again,  let  x  be  the  number  of  kilogram-meters  in  1  foot-pound, 

then 

X  kg.  m.  =  1  ft.  lb., 
hence 

^  ^  Ik     ft.  ^oi3g257 
kg.    m. 

i.  e.  1  foot-pound  =  0.138  257  kilogram-meters. 

Finally,  1  foot-pound  =  g  foot-poundals  (Art.  179);  hence  1  foot- 
pound =  1.356  X  10^  ergs,  and  1  erg  =  7.3730  X  10~^  foot-pounds,  if 
g  =  981. 

264.  Exercises. 

(1)  A  joule  being  defined  as  10^  ergs,  show  that  1  foot-pound  = 
1.356  joules,  and  that  1  joule  is  about  3/4  foot-pound. 

(2)  Show  that  a  kilogram-meter  is  nearly  10^  ergs. 

(3)  "WTiat  is  the  work  done  against  gravity  in  raising  300  lbs.  through 
a  height  of  25  ft. :   (a)  in  foot-pounds,  (5)  in  ergs? 


265.]  THEORY   OF  ATTRACTIVE   FORCES  203 

(4)  Find  the  work  done  against  friction  in  moving  a  car  weighing 
3  tons  through  a  distance  of  50  yards  on  a  level  road,  the  coefficient 
of  friction  being  0.02. 

(5)  A  mass  of  12  lbs.  slides  down  a  smooth  plane  inclined  at  an 
angle  of  30°  to  the  horizon,  through  a  distance  of  25  ft.;  what  is  the 
work  done  by  gravity? 

265.  The  work  of  a  variable  force  i^  in  a  very  small  dis- 
placement PP'  =  8s  is  defined  (like  that  of  a  constant  force 
in  any  displacement,  Art.  261)  as  the  product  of  5s  into  the 
projection  F  cosi^  of  F  (at  P)  on  8s: 

8W  ^  F  cos^  8s  =  F-8s  =  X8x  +  Y8y  +  Z8z. 

This  expression  is  often  called  the  virtual  work  of  F  in 
the  virtual  displacement  5s,  the  term  virtual  and  the  letter  5 
meaning  that  the  displacement  is  arbitrary  and  not  neces- 
sarily the  actual  displacement  along  the  path  of  the  particle. 

But  it  should  be  carefully  observed  that  even  if  the  dis- 
placement 5s  were  taken  along  the  actual  path  we  do  not  in 
general  have  in  the  limit 

dW       „        , 

—^  =  /<  cos;/'; 
as 

i.  e.  the  s-component  of  the  force  is  not  necessarily  an  exact 
derivative. 

The  work  done  by  the  variable  force  F  as  the  particle 
on  which  it  acts  is  moved  along  an  arbitrary  curve  from  Po 
to  any  position  P  is  written 

W  =  lim  SP  cosiA  5s  =   ^F  cos;/'  ds  =  f  V  •  ds 

hs=Q  'JPo  '^Po 

=  S^{Xdx  +  Ydy  +  Zdz). 

This  integral  can  in  general  not  be  evaluated  unless  the  path 
of  the  particle  from  Po  to  P  is  known;  and  it  has  in  general 


204  STATICS  [266. 

different  values  for  different  paths  between  these  points. 
But  we  have  seen  (Arts.  257,  258)  that  for  a  particle  m 
in  a  field  of  Newtonian  attraction  the  component  of  the 
resultant  attraction  in  any  direction  s  is  the  s-derivative  of 
the  potential:  As  =  dU/ds.  Hence,  multiplying  by  m,  we 
have  in  this  case  for  the  virtual  work: 

""''"■'  8W  =  mAs8s  =  mdU. 

It  follows  that  the  work  done  on  the  particle  m  by  the  New- 
tonian attraction,  as  it  is  moved  from  Pq  to  P  along  any 
path,  is 

ui  H  "^  =  mfl'sU  =  m{U  -  Uo), 

where  Uo  is  the  value  of  U  at  Pq.  Hence  the  work  of  attrac- 
tion is  independent  of  the  -path;  it  is  m  times  the  difference  of 
'potential  at  P  and  Pq]  it  is  zero  in  any  closed-  path. 

•  More  generally,  whenever  the  force  F  is  conservative  (Art. 
255)  so  that  it  possesses  a  one-valued  force-function,  i.  e.  a 
function  U{x,  y,  z)  such  that  dU/dx,  dll/dy,  dU/dz  are  the 
rectangular  components  of  F,  the  projection  of  F  on  any 
direction  s  will  be  the  s-derivative  of  U,  and  hence  the  work 
of  F  is  independent  of  the  path. 

266.  For  a  particle  in  equilibrium,  since  the  resultant  force 
F  is  zero,  it  follows  that  the  virtual  work  bW  =  F  cos;/'  bs  is 
zero  whatever  the  displacement  5s.  And  conversely,  if  the 
virtual  work  is  zero  whatever  5s,  or  more  exactly,  if  the 
virtual  work  is  small  of  an  order  higher  than  that  of  5s  for 
every  sufficiently  small  5s,  the  resultant  force  F  must  be  zero, 
i.  e.  the  particle  is  in  equilibrium. 

The  virtual  work  is  zero  for  every  5s  if  it  is  zero  for  any 
three  non-complanar  displacements. 
.    'Using  rectangular    co-ordinates  we   have   8W  =  X8x -{- 


268.]  THEORY   OF   ATTRACTIVE   FORCES  205 

Ydy  +  Z8z;  hence  8W  =  0  when  X  =  0,  F  =  0,  Z  =  0; 
and  conversely,  if  8W  =  0  for  a  virtual,  i.  e.  arbitrary,  dis- 
placement, we  have  owing  to  the  independence  of  8x,  8y,  8z'. 
m  =  0. 

The  proposition  that  the  vanishing  of  the  virtual  work  (apart 
from  terms  of  a  higher  order)  is  a  necessary  and  sufficient 
condition  of  equilibrium  for  a  particle  is  known  as  the  principle 
of  virtual  work  for  the  particle. 

267.  In  the  particular  case  of  a  particle  in  a  field  of  con- 
servative forces  whose  force-furiction  is  U,  the  condition  of 
equilibrium  assumes  the  form 

ds 
for  any  ds  •  or,  with  reference  to  rectangular  axes : 

dx  dy  dz 

Now  these  are  necessary  conditions  for  a  maximum  or 
minimum  of  U.  Hence  the  positions  of  equilibrium  of  a 
particle  under  conservative  forces  are  found  by  determining 
the  maxima  and  minima  of  the  force-function  or  potential. 

It  can  be  shown  that  a  minimum  of  U  corresponds  to 
stable,  a  maximum  to  unstable,  equilibrium. 

268.  The  principle  of  virtual  work,  proved  above  only  for 
the  single  free  particle,  has  a  far  wider  field  of  application.  It 
can  be  shown  that  for  any  system  of  particles  or  rigid  bodies, 
subject  to  any  constraints,  expressible  by  equations  (not  in- 
equalities) a7ul  not  involving  friction,  the  vanishing  of  the  virtual 
work  (apart  from  terms  of  higher  order)  for  any  displacement 
compatible  with  the  constraints  is  a  necessary  and  sufficient 
condition  of  equilibrium. 


206  STATICS  [268. 

If  in  the  expression  of  the  virtual  work  8W  =  F  cosi/'  8s 
we  replace  8s  by  {8s/8t)8t  we  can  regard  8s/8t  as  a  velocity. 
This  is  the  reason  why  the  principle  of  virtual  work  is  often 
called  the  principle  of  virtual  velocities. 


PART  III:  KINETICS. 


CHAPTER  XIII. 
MOTION  OF   A  FREE   PARTICLE. 

1.  The  equations  of  motion. 

269.  Let  a  particle  of  mass  m  be  acted  upon  by  any  number 
of  forces;  as  these  forces  are  concurrent  they  are  equivalent 
to  a  single  resultant  R  (Art.  190).  The  definition  of  force 
(Art.  171)  then  gives  for  the  acceleration  j  the  fundamental 

equation  of  motion 

mi  =  R.  (1) 

The  mass  m  being  regarded  as  a  positive  constant  the  equa- 
tion shows  that  the  vectors  j  and  R  have  the  same  direction 
and  sense. 

The  vector  equation  (1)  assumes  various  forms  according 
to  the  method  selected  for  resolving  j  and  R  into  components. 

If  the  motion  be  referred  to  fixed  rectangular  axes,  (1)  is 
replaced  by  the  three  equations  (Art.  53) : 

mx  =  X,     my  =  Y,     mz  =  Z,  (2) 

X,  Y,  Z  being  the  components  of  R  along  Ox,  Oy,  Oz. 
If  polar  co-ordinates  r,  6,  cp  are  used  we  have  (Art.  56, 

Ex.  9) : 

m(r  —  rd^  —  r  mi'^d  (f"^)  =  Rr, 

m(rB  +  2fd  -  r  sin0  cos0  <p^)  =  Re,  (3) 

m(r  sin0  ip  -\-  2  sin0  f<p  +  2r  cosd  d<p)  =  R^, 
207 


208  KINETICS  [270. 

where  Rr,  Re,  R<t>  are  the  components  of  R  along  the  radius 
vector,  at  right  angles  to  the  radius  vector  in  the  meridian 
plane,  and  at  right  angles  to  this  plane. 

Finally,  resolving  along  the  tangent,  normal,  and  bi- 
normal  to  the  path  we  have  (Art.  51) : 

mv  =  ms  =  Rt,     m  —  =  i?„,     0  —  Rb.  (4) 

P 

In  the  case  of  plane  motion  the  equations  (2),  (3),  (4) 
reduce  to  the  first  two,  with  ^  =  0  in  (3);  in  the  case  of 
rectilinear  motion  the  first  equation  of  (2)  or  (4)  suffices. 

270.  If  the  components  X,  Y,  Z  were  given  as  functions 
of  the  time  t  alone,  each  of  the  three  equations  (2)  could  be 
integrated  separately.  In  general,  however,  these  com- 
ponents will  be  functions  of  the  co-ordinates,  and  perhaps 
also  of  the  velocity  and  of  the  time.  No  general  rules  can 
be  given  for  integrating  the  equations  in  this  case.  By  com- 
bining the  equations  (2)  in  such  a  way  as  to  produce  exact 
derivatives  in  the  resulting  equation,  it  is  sometimes  possible 
to  effect  an  integration.  Two  methods  of  this  kind  have 
been  indicated  for  the  case  of  two  dimensions  in  a  particular 
example  in  Kinematics,  Arts.  102-104.  We  now  proceed  to 
study  these  principles  from  a  more  general  point  of  view,  and 
to  point  out  the  physical  meaning  of  the  expressions  involved. 

271.  The  Principle  of  Kinetic  Energy  and  Work.  Let  us 
combine  the  equations  of  motion  (2)  by  multiplying  them 
by  X,  y,  z,  respectively,  and  then  adding.  As  xx  is  the  time 
derivative  of  ^x-,  the  left-hand  member  of  the  resulting 
equation  will  be  the  ^-derivative  of  ^m{x-  -\- ij^  -\-  z^)  =  ^ww^, 
i.  e.  of  the  kinetic  energy  of  the  particle  (Art.  181).  We  find 
therefore 

j^^mv'-  =  Xx+  Yy  +  Zz. 


272.]  MOTION  OF  A  FREE  PARTICLE  209 

Hence,  integrating  from  any  point  Po  of  the  path  where 
z;  =  i^o  to  any  point  P  we  obtain : 

^mv^  -  ^vo'  =  fl^iXdx  +  Ydy  +  Zdz).  (5) 

The  left-hand  member  represents  the  increase  in  the  kinetic 
energy  of  the  particle;  the  right-hand  member  represents 
the  work  done  by  the  resultant  force  R,  since  its  work  is 
equal  to  the  sum  of  the  works  of  its  components  X,  Y,  Z 
(Art.  261).  Equation  (5)  states,  therefore,  that  the  amount 
hij  which  the  kinetic  energy  increases,  as  the  particle  passes  from 
Po  to  P,  is  equal  to  the  work  done  by  the  resultant  force  R  on 
the  particlt. 

272.  The  principle  of  kinetic  energy  and  work  can  also  be 
deduced  from  the  former  of  the  two  equations  (4).  Multiply- 
ing this  equation  by  w  =  dsjdt,  we  have 

dihrnv"^)        T^  ds       „         ,ds 

hence,  integrating  as  in  Art.  271: 

^iv^  —  ^Vo^  =  J  R  cosi/'  ds,  (5') 

where  \p  is  the  angle  made  by  the  force  R  with  the  tangent 
to  the  path. 

The  integrand  in  (5)  or  (5'),  i-  e.  the  expression 

R  cofi\p  ds  =  R-ds  =  Xdx  +  Ydy  +  Zdz, 

is  called  the  elementary  work.  It  is  the  value  of  the  virtual 
work  (Art.  265)  when  the  displacement  5s  is  taken  infini- 
tesimal and  along  the  actual  path. 

As  explained   in  Art.   265,   the  evaluation  of  the  work 
integral  in  general  requires  a  knowledge  of  the  path.     As  in 
many  problems  the  path  is  not  known  ])eforehand,  but  is 
15 


210  KINETICS  [273. 

one  of  the  things  to  be  determined,  it  is  very  important  to 
notice  that  in  the  case  of  conservative  forces  (Art.  255)  the 
work  integral  has  a  value  independent  of  the  path  (Art.  265). 
In  this  case,  denoting  the  force-function,  or  potential,  by  U, 
we  have 

f^iXdx  +  Ydy  +  Zdz)  =  f^^dU  =  U  -  Uo, 

so  that  the  equation  (5)  or  (5')  becomes 

^nv"  —  ^Vq^  =  U  —  Uo-  (6) 

Hence  in  the  case  of  conservative  forces  the  principle  of 
kinetic  energy  and  work  at  once  gives  a  first  integral  of  the 
equations  of  motion. 

273.  The  negative  of  the  force-function,  say 

V=  -U, 

is  called  the  potential  energy.  If  this  quantity  be  intro- 
duced and  the  kinetic  energy  be  denoted  by  T,  the  equation 
(6)  assumes  the  form 

T+V=To+  Vo,  (6') 

which  expresses  the  principle  of  the  conservation  of  energy 
for  a  particle:  the  total  energy,  i.  e.  the  sum  of  the  kinetic  and 
potential  energies,  remains  constant  throughout  the  motion  ij 
the  forces  are  conservative.  In  other  words,  whatever  is  gained 
in  kinetic  energy  is  lost  in  potential  energy,  and  vice  versa. 

274.  The  physical  idea  to  which  the  term  potential  energy  is  due 
can  perhaps  best  be  explained  by  considering  the  Newtonian  attraction 
between  two  particles  m,  m'.  We  think  of  the  attracting  particle  to' 
as  generating  a  field.  ^\Tierever  in  this  field  a  particle  m  be  placed 
(say,  Nvath  zero  velocity),  it  will  become  subject  to  the  attraction  A  of  m' 
and  move  toward  m'  with  increasing  velocity,  thus  acquiring  kinetic 
energy;  at  the  same  time  the  force  A  does  an  amount  of  work  on  ni  which 
is  exactly  equivalent  to  the  kinetic  energy  gained  by  m.     It  follows  that, 


276.]  MOTION   OF  A   FREE   PARTICLE  211 

the  farther  away  from  m'  the  particle  m  is  placed,  initially,  the  greater 
will  be  the  amount  of  work  that  m'  can  do  upon  it.  It  is  this  "poten- 
tiality" for  doing  work,  due  to  the  distance  of  m  from  m',  which  is 
denoted  as  energy  of  -position,  or  potential  energy.  The  equation  (6), 
or  the  equation  (6')  which  differs  from  (6)  merely  in  notation,  shows 
that  what  the  particle  m  in  moving  toward  m'  gains  in  kinetic  energy  it 
loses  in  potential  energy  so  that  the  sum  of  kinetic  and  potential  energy 
always  remains  constant. 

275.  The  conditions  for  the  existence  of  a  force-function 
are  (Art.  255) : 

X  =  ^J1       Y  =  ^^      Z  ^  — 

dx  '  djj  '  dz  ' 

Differentiating  the  second  equation  with  respect  to  z,  the 
third  with  respect  to  y  we  find 

dY  _  dnj_      dZ  _  dHJ_ 
dz        dzdy '     dy        dydz ' 

whence  dY/dz  =  dZ/dy.  Proceeding  in  the  same  way  with 
the  other  two  pairs  of  equations  we  find : 

dY  _dZ      dZ  _  dX      dX  _  dY_ 

dz        dy'     dx         dz  '      dy         dx  ' 

These  relations  which  are  necessary  and  sufficient  for  the 
existence  of  a  force-function  U  furnish  a  simple  criterion  for 
recognizing  whether  the  given  forces  are  conservative. 

276.  The  principle  of  the  conservation  of  energy,  i.  e. 
of  the  constancy  of  the  sum  of  kinetic  and  potential  energy, 
has  been  proved  mathematically  in  the  preceding  articles 
for  a  very  particular  case,  viz.  for  the  motion  of  a  particle 
under  conservative  forces. 

By  a  generalization  as  bold  and  far-reaching  as  was  New- 
ton's extension  of  the  property  of  mutual  attraction  to 
all  matter  (Art.  244),  modern  physics  has  been  led  to  the 
assumption  that  .work  and  energy  are  quantities  which  can 


212  KINETICS  [277. 

never  he  destroyed,  but  can  be  transformed  in  a  variety  of 
ways.  This  assumption,  the  general  principle  of  the  con- 
servation of  energy,  while  fully  borne  out  so  far  by  the 
results  deduced  from  it,  is  of  course  not  capable  of  math- 
ematical proof.  Indeed,  it  may  be  said  that  in  defining 
the  various  forms  of  energy,  such  as  heat,  chemical  energy, 
radio-activity,  etc.,  the  definitions  are  so  formulated  as  to 
conform  to  this  principle;  it  has  always  been  found  possible 
to  do  this.  The  general  principle  of  the  conservation  of 
energy  cannot  be  fully  discussed  here,  since  this  would 
require  a  study  of  all  the  forms  of  energy  known  to  physics. 

277.  In  its  application  to  machines,  the  principle  states  that  the 
total  work  W  suppUed  to  a  machine  in  a  given  time  by  the  agent,  or 
motor,  driving  it  (such  as  animal  force,  the  expansive  force  of  steam, 
the  pressure  of  the  wind,  the  impact  of  water,  etc.)  is  equal  to  the  sum 
of  the  useful  work  Wv,  done  by  the  machine  in  the  same  time  and  the 
so-called  lost,  or  wasteful,  work  Wu-  spent  in  overcoming  friction  and 
other  passive  resistances  of  the  machine: 

W    =    Wu    +    Wu: 

While  W  and  Wu  can  be  determined  with  considerable  accuracy, 
it  is  difficult  to  determine  Ww  directly  with  equal  precision;  but  it  is 
found  that  the  more  accurately  in  any  given  machine  Ww  is  determined, 
the  more  nearly  will  the  above  equation  be  found  satisfied.  This 
serves  as  a  verification  of  the  principle  of  the  conservation  of  energy  in 
its  application  to  machines.  The  ratio  W„/W  of  the  useful  work  to 
the  total  work  is  called  the  efficiency  of  the  machine.  The  term 
modulus  is  sometimes  used  for  efficiency. 

278.  The  time-rate  at  ivhich  irork  is  performed  by  a  force  has  received 
a  special  name,  power  or  activity.  The  source  from  which  the  force 
for  doing  useful  work  is  derived  is  commonly  called  the  agent,  or  motor; 
and  it  is  customary  to  speak  of  the  power  of  an  agent,  this  meaning 
the  rate  at  which  the  agent  is  capable  of  supphang  work. 

The  dimensions  of  power  are  evidently  ML-T~^.  Tlie  unit  of  power 
is  the  power  of  an  agent  that  does  unit  work  in  unit  time.     Hence, 


279.]  MOTION   OF  A   FREE   PARTICLE  213 

in  the  scientific  system,  it  is  the  power  of  an  agent  doing  one  erg  per 
second  in  the  C.G.S.  system,  and  one  foot-poundal  per  second  in  the 
F.P.S.  system.  As,  however,  the  idea  of  power  is  of  importance  mainly 
in  engineering  practice,  power  is  usually  measm'ed  in  gravitation  units. 
In  this  case,  the  unit  of  power  is  the  power  of  an  agent  doing  one  foot- 
pound per  second  in  the  F.P.S.  system,  and  one  kilogram-meter  in  the 
metric  system. 

A  larger  unit  is  frequently  found  more  convenient.  For  this  reason, 
the  name  horse-power  (H.P.)  is  given  to  the  power  of  doing  .550  foot- 
pounds of  work  per  second,  or  550  X  60  =  33,000  foot-pounds  per 
minute. 

279.  The  principle  of  angular  momentum  or  of  areas. 

By  multiplying  the  first  of  the  equations  of  motion  (2), 
Art.  269,  by  y,  the  second  by  x,  and  then  subtracting  the 
first  from  tiie  second  we  obtain  the  equation 

m{xij  —  yx)  =  xY  —  yX, 

or  since  the  left-hand  member  is  the  time-derivative  of 
m{xy  —  yx): 

jm^xy  —  yx)  =  xY  —  yX. 

Here  the  right-hand  member  is  the  moment  of  the  resultant 
force  R  about  the  axis  Oz  (Art.  229)  while,  on  the  left,  the 
quantity  x  •  my  —  y  •  mx  is  the  moment  about  the  same  axis 
of  the  inomentum  mv  whose  components  are  mx,  my,  mz 
(Art.  168).  This  moment  of  momentum  m{xy  —  yx)  is 
also  called  angular  momentum. 

As  any  line  might  have  been  chosen  as  axis  Oz,  our  equa- 
tion expresses  the  proposition:  In  the  motion  of  a  ^article, 
the  time-rate  of  change  of  the  angular  momentum  about  any 
line  is  equal  to  the  moment  of  the  resultant  force  about  the  same 
line. 

Applying  this  result  to  each  of  the  axes  of  reference  we  find : 


214 


KINETICS 


[280. 


-^^miyz  -  zy)  =  yZ  -  zY, 


m{zx  —  xz)  =  zX  —  xZf 
m(xy  —  yx)  =  xY  —  yX. 


(8) 


These  equations  express  the  principle  of  angular  momentum 
or  of  areas. 

280.  To  interpret  these  equations  geometrically  consider 
first  the  right-hand  members  which  are  the  moments  of 
the  resultant  force  R  about  the  axes.  The  vector  PA  =  R 
(Fig.  70)  forms  with  the  origin  0  a  triangle  whose  area  is 


Fig.  70. 


ome-half  the  moment  of  R  about  0;  let  us  represent  this 
moment,  which  is  the  cross-product  of  the  radius  vector 
OP  =  r  and  the  force-vector  PA  =  R,  hj  a  vector  H  per- 
pendicular to  the  plane  of  the  triangle  OPA  (comp.  Arts. 
199  and  119): 

H  =  rXR, 


281.]  MOTION   OF  A   FREE   PARTICLE  215 

the  length  of  this  vector  H  being  equal  to  twice  the  area  OP  A. 
The  projection  of  the  triangle  OP  A  on  the  a;y-plane  has 
the  area  ^{xY  —  yX)  since  the  vertices  of  this  projection 
have  the  co-ordinates  (0,  0),  {x,  y),  and  {x-\-  X,  y -{-  Y)- 
hence  the  right-hand  members  of  (8)  are  the  components 
Hjc,  Hy,  Hz  of  the  vector  H. 

Next  consider  in  the  same  way  the  momentum-vector 
mv  =  PB;  it  forms  with  0  a  triangle  OPB  whose  area  is 
one-half  the  moment  of  momentum  about  0.  We  can 
represent  this  moment  of  momentum,  or  angular  momentum, 
by  a  vector  h,  perpendicular  to  the  plane  OPB,  and  of  a 
length  equal  to  twice  the  area  of  the  triangle  OPB;  the 
vector  h  is  then  the  cross-product  of  r  =  OP  and  mv  =  PB : 

h  =  r  X  mv. 

The  components  of  angular  momentum  m(yz  —  zy), 
m(zx  —  xz),  m(xy  —  yx)  are  the  components  hx,  hy,  hz 
of  the  vector  h. 

The  equations  (8)  can  therefore  be  written  in  the  form 

dT-^-    rfT-^-    df^^"  ^^^ 

and  these  equations  can  be  combined  into  the  single  vector 
equation 

dh      „ 

which  means  that  the  geometrical  increment  of  the  vector 
h,  divided  by  A^,  gives  in  the  limit  the  vector  H;  i.  e.  the 
(geometrical)  time-rate  of  change  of  the  angular-momentum 
vector  is  equal  to  the  moment-vector  of  the  residtant  force. 

281.  If  instead  of  the  momentum- vector  mv  we  consider 
the  velocity-vector  v,  its  moment  about  0  would  be  repre- 
sented by  the  vector  {l/m)h,  whose  components  are  yz  —  zy, 


216  laNETICS  [282. 

zx  —  xz,  xy  —  yx.  These  quantities  are  (Art.  47)  equal  to 
twice  the- sectorial  velocities  about  the  axes  while  the  vector 
{\/ni)h  represents  twice  the  sectorial  velocity  of  the  particle 
about  0.     This  explains  the  name  principle  of  areas. 

282.  If,  in  particular,  the  resultant  force  R  is  central,  i.  e. 
such  as  to  pass  always  through  a  fixed  point,  then,  for  this 
point  as  origin,  the  right-hand  members  of  the  equations  (8) 
are  zero,  and  we  find  at  once  the  first  integrals  of  the  equa- 
tions of  motion  (2) : 

m{yz  —  zy)  =  hi,     m{zx  —  xz)  =h2,     m(xy  —  yx)  =  h,      (9) 

where  hi,  ho,  hs  are  constants. 

Thus,  in  the  motion  of  a  particle  in  the  field  of  a  central 
force,  the  angular  momentum,  and  hence  the  sectorial  veloc- 
ity, about  any  axis  through  the  center  is  constant. 

If  the  resultant  force  always  intersects  a  fixed  line,  the 
angular  momentum,  and  hence  the  sectorial  velocity,  about 
this  line  as  axis  remains  constant. 

These  propositions  are  often  referred  to  as  the  principle  of 
the  conservation  of  angular  momentum  or  of  areas. 

It  may  be  noted  that  the  equations  (9),  multiplied  by 
X,  y,  z  and  added  give, 

hix  +  h^iy  +  hzz  =  0; 

this  shows  that  the  particle  moves  in  a  plane  passing  through 

the  center  of  force,  as  is  otherwise  evident. 

283.  Exercise. 

In  the  case  of  plane  motion,  if  the  plane  be  taken  as  the  xy-plane, 
the  principle  of  areas  is  expressed  by  the  third  of  the  equations  (8).  If 
the  perpendicular  from  the  origin  0  to  the  tangent  at  P  be  denoted  by  p 
(comp.  Art.  100),  this  equation  can  be  written  in  the  form  d{mpv)/dt  = 
xY  ~  yX.  Show  that  the  two  terms  mpdv/dt  and  mvdp/dt  of  the  left- 
hand  member  represent  the  moments  of  the  tangential  and  normal 
components  of  the  resultant  force  R,  respectively. 


285.1 


MOTION   OF  A   FREE   PARTICLE  217 


2.  Examples  of  rectilinear  motion. 
284.  Free  Oscillations.  As  an  example  of  rectilinear  mo- 
tion consider  the  motion  of  a  particle  of  mass  m  under  a  force 
directly  proportional  to  the  distance  OP  =  s  of  the  particle 
from  a  fixed  point  0.  If  the  force  is  attractive,  i.  e.  directed 
toward  the  point  0  and  if  the  initial  velocity  passes  through 
0  or  is  zero  so  that  the  motion  is  rectilinear,  the  single 
equation  of  motion  is 

m's  =  —  MK^s,  (10) 

and  the  motion  (see  Arts.  26,  27,  71)  is  a  simple  harmonic 
oscillation  or  vibration  about  the  point  0  as  center.  This 
point  0,  at  which  the  force  R  ^  —  ynn^s  is  zero,  is  therefore 
a  position  of  ecjuilibrium  for  the  particle. 

The  potential  energy  V  due  to  the  force  R  =  —  mKrs  is,  by 
Art.  273, 

F  =  —    I  Rds  =  mK~  I   sds  =  ^vikts'^  +  C. 

Hence  the  principle  of  the  conservation  of  energy  gives 
y2  _j_  ^2^2  =  const. 

If  the  initial  velocity  be  zero  for  s  =  So,  we  have 

y  =   =f:  /C  V.S'o"  —  s^. 

285.  As  in  the  applications  the  moving  particle  m  is  generally  subject 
to  the  constant  force  of  gravity,  it  is  important  to  notice  that  the  intro- 
duction of  a  constant  force  F  along  the  line  of  motion  does  not  essentially 
change  the  character  of  the  motion.     For,  the  equation  of  motion 

ms  =  —  m.K^s  -\-  F  =  —  niK^  {  s  —       „] 
reduces,  with  s  —  FlmK^  =  x,  to 

mx  =  —  niK^x, 
which  agrees  in  form  with  (10).     The  only  change  in  the  results  is  that 


218  KINETICS  [286. 

the  center  of  the  oscillations,  i.  e.  the  position  of  equilibrium  of  the 
particle  m,  is  not  the  point  0,  but  a  point  at  the  distance  e  =  F/mK^ 
from  0. 

286.  Forces  proportional  to  a  distance,  or  length,  are  directly  ob- 
served in  the  stretching  of  so-called  elastic  materials.  Thus,  a  homo- 
geneous straight  steel  wire  when  suspended  vertically  from  one  end 
and  weighted  at  the  other  end  is  found  to  stretch;  and  careful  measure- 
ments have  shown  that  the  extension,  or  change  of  length,  is  directly 
proportional  to  the  weight  apphed  (the  weight  of  the  wire  itself  being 
assumed,  for  the  sake  of  simplicity,  as  very  small  in  comparison  with 
the  load  applied).  Conversely,  the  tension,  or  elastic  stress,  of  the  wire 
is  proportional  to  the  extension  produced.  INIoreover,  when  the  weight 
is  removed  the  wire  is  found  to  contract  to  its  original  length. 

This  physical  law,  known  as  Hooke's  law  of  clastic  stress,  holds  only 
within  certain  limits.  If  the  weight  exceeds  a  certain  limiting  value,  the 
extension  is  no  longer  proportional  to  the  weight,  and  after  removing 
the  weight,  the  wire  does  not  regain  its  original  length,  but  is  found  to 
have  acquired  a  permanent  set,  or  lengthening;  it  is  said  in  this  case 
that  the  elastic  limits  ha,ve  been  exceeded. 

Materials  for  which  Hooke's  law  holds  exactly  witliin  certain  limits 
of  tension  and  extension  are  called  perfectly  elastic.  Strictly  speaking, 
such  materials  probably  do  not  exist;  but  many  materials  follow  Hooke's 
law  very  closely  within  proper  limits.  Thus,  elastic  strings,  such  as 
rubber  bands,  and  spiral  steel  springs  show  these  phenomena  very 
clearly  on  account  of  the  large  extensions  allowable  within  the  elastic 
limits. 

287.  The  elastic  constant  mn.^.  Let  an  elastic  string  whose  natural 
length  is  I  assume  the  length  I  +  x  when  the  tension  is  F,  so  that  accord- 
ing to  Hooke's  law, 

F  =  -  mn^x. 

To  determine  the  factor  of  proportionaHty  ?nK^  for  a  given  string,  we 
may  observe  the  length  h  assumed  by  the  string  under  a  known  ten- 
sion, e.  g.  the  tension—  mig  produced  by  suspending  a  given  mass  nii 
from  the  string  (the  weight  of  the  string  itself  being  neglected). 

We  then  have 

—  mig  =  —  rnK^ili  —  l), 
whence 


289.]  MOTION  OF  A  FREE  PARTICLE  219 

and 

288.     Let  the  same  string  be  placed  on  a  smooth  horizontal  table, 
one  end  being  fixed  at  a  point  0  (Fig.  71),  while  a  particle  of  mass  m  is 

|<- j^-aji ^ 

oU 1 


W/MZ/ymMSm 


Fig.  7L 

attached  to  the  other  end.  Stretch  the  string  to  a  length  OPo  =  i  +  Xo 
(within  the  limits  of  elasticity)  and  let  go;  the  particle  m.  will  move 
under  the  action  of  the  tension  F  alone,  its  weight  being  balanced  by 
the  reaction  of  the  table.     The  equation  of  motion  is 

vix  =   —  ^ X, 

the  distance  QP  =  x  being  counted  from  the  point  Q  at  the  distance 
OQ  =  I  from  the  fixed  point  0,     Putting  again  (Art.  287) 


\mih  -  0  ' 
and  integrating,  we  find 

X  =  Ci  coskI  +  C2  smd, 
whence 

V  =  X  =  —  KCi  sind  +  kCz  coskL 

As  X  =  Xo  and  v  =  0  ior  t  =  0,  we  have  Ci  =  xo,  ct  =  0;  hence 
X  =  Xo  cosd,     V  =  —KXo  sind. 

It  should  be  noticed  that  these  equations  hold  only  as  long  as  the 
string  is  actually  stretched,  i.  e.  as  long  as  x  >  0.  The  subsequent 
motion  is,  however,  easily  determined  from  the  velocity  for  x  =  0. 

289.  It  was  assumed,  in  the  preceding  article,  that  the  particle  m  is 
let  go  from  its  initial  position  Po  with  zero  velocity.  This  can  be 
brought  about  by  pulling  the  particle  from  Q  to  Po  with  a  gradually 
increasing  force  which  at  any  point  P  is  just  equal  and  opposite  to  the 


220  laNETICS  1290. 

corresponding  elastic  tension,  or  stress,  P  =  friK^x.  The  work  thus  done 
against  the  tension,  i.  e.  in  stretching  or  straining  the  string,  is  stored  in 
the  particle  m  as  potential  energy,  or  strain  energy,  V.  To  find  ita 
amount,  observe  that,  as  the  particle  7n  is  pulled  through  the  short  dis- 
tance A.r,  the  work  of  the  force  is  =  7«AAx;  this  being  the  potential 
energy  AV  gained  in  the  distance  Ax,  we  have  AT  =  wk^xAx;  hence 

Vo  =    I     niK-xdz  =  Ijuk-xo'. 

fJ  0 

Thus,  in  the  initial  position  Po  the  particle  m  possesses  this  potential 
energy,  but  no  kinetic  energy.  During  its  motion  from  Po  to  Q,  the 
particle  gains  kinetic  energy  and  loses  potential  energy.  At  any  inter- 
mediate point  P,  for  which  QP  =  x,  the  kinetic  energy  is  T  =  Imv^, 
while  the  potential  energy  is  V  =  im/c^x^.  By  the  principle  of  the  con- 
servation of  energy  (Art.  273),  the  sum  of  these  two  quantities,  the 
so-called  total  energy,  E,  remains  constant  as  long  as  no  other  forces 
besides  the  elastic  stress  act  on  the  particle: 

^mv-  +  huK-x^  =  const. 

The  value  of  the  constant  is  =  hnK-xo',  since  this  is  the  total  energy 
at  Po;  hence, 

I'-    +   K-X-    =    K-Xlf. 

(Comp.  Art.  284).  This  relation  also  follows  from  the  values  of  x  and 
V  given  in  Art.  2SS,  upon  eliminating  /. 

When  the  particle  arrives  at  the  position  of  equilibrium  Q,  the 
potential,  or  strain,  energy  has  been  consumed,  having  been  converted 
completely  into  kinetic  energy. 

290.  Exercises. 

(1)  In  the  problem  of  Art.  2SS  let  the  string  be  a  rubber  band  whose 
natural  length  of  1  ft.  is  increased  3  in.  when  a  weight  of  4  oz.  is  suspended 
from  it;  determine  the  motion  of  a  1-oz.  particle  attached  to  one  end, 
the  band  being  initially  stretched  to  a  length  of  1)^  ft.;  find  (a)  the 
greatest  tension  of  the  band,  (b)  the  greatest  velocitj^  of  the  particle, 
(c)  the  period,  (d)  the  work  done  by  the  tension  in  a  quarter  oscillation. 

(2)  Discuss  the  effect  of  friction,  of  coefficient  fi,  in  the  problem  of 
Art.  28<S. 

(3)  The  length  OQ  =  Z  of  an  elastic  string  is  increased  to  OQi  =  h  = 


291.]  MOTION   OF  A  FREE   PARTICLE  221 

I  +  e  a  a.  mass  wi  is  suspended  from  its  lower  end,  the  upper  end  0 
being  fixed  (Fig.  72).     The  mass  m  is  pulled  down  to  the  distance 
QiFo  =  Xo  from  the  position  of  equilibrium  Qi  and  then  released.     Prove 
the  following  results:  With  Qi  as  origin  the  equation  of 
motion  of  ?)i  is 


X  =  —  K^x,  where  k  =-«(?- 
\e 
whence 

X  =  Xo  coskI,  V  =  —  KXo  siuKt. 

If  Xo  <  e,  the  tension  never  vanishes,  and  m  performs 
isochronous  oscillations  of  period  2irVe/g,  the  period  be- 
ing the  same  as  for  the  small  oscillations  of  a  pendulum 
of   length  e.     If    xo  >  e,  the   tension   vanishes  for  x  = 

—  e,   i.   e.   at   Q;   the   velocity    at    this    point   is    vi  = 

—  kV xa'  —  e^,     and    the    particle    rises    to    the     height 
h  =  (xo^  —  e2)/2e  above   Q.     The    total    time  of    one  up        pj^.  72 
and  down  motion  is 

2V^g[hTr  +  sin-Ke/xo)  +  Vjxo/c)^  -  1]. 

(4)  How  is  the  motion  of  Ex.  (3)  modified  if  the  elastic  string  be 
replaced  by  a  spiral  spring  suspended  vertically  from  one  end?  Assume 
the  resistance  of  the  spring  to  compression  equal  to  its  resistance  to 
extension. 

(5)  The  particle  in  Ex.  (3)  is  let  fall  from  a  height  h  above  Q;  deter- 
mine the  greatest  extension  of  the  string. 

(6)  An  elastic  string  whose  natural  length  is  I  is  suspended  from  a 
fixed  point.  A  mass  nii  attached  to  its  lower  end  stretches  it  to  a  length 
h;  another  mass  m2  stretches  it  to  a  length  h.  If  both  these  masses  be 
attached  and  then  the  mass  ???2  be  cut  off,  what  will  be  the  motion  of 
nil? 

(7)  If  a  straight  smooth  hole  be  bored  through  the  earth,  connecting 
any  two  points  A,  B  on  the  surface,  in  what  time  would  a  particle  slide 
from  A  to  B?  The  attraction  in  the  interior  is  directly  proportional 
to  the  distance  from  the  center  of  the  earth. 

291.  Resistance  of  a  Medium.  It  is  known  from  obser- 
vation that  the  velocity  y  of  a  rigid  body  moving  in  a  liquid 
or  gas  is  continually  diminished,  the  medium  apparently  exert- 


222  KINETICS  [291. 

ing  on  the  body  a  retarding  force  which  is  called  the  resistance 
of  the  medium.  This  force  F  is  found  to  be  roughly  propor- 
tional to  the  density  p  of  the  medium,  the  greatest  cross- 
section  A  of  the  body  (at  right  angles  to  the  velocity  v),  and 
generally,  at  least  for  large'*velocities,  to  the  square  of  the 

velocity  v. 

F  =  kpAv"", 

where  A;  is  a  coefficient  depending  on  the  shape  and  physical 

condition  of  the  surface  of  the  body. 

This  expression  for  the  resistance  F  can  be  made  plausible  by  the 
following  consideration.  As  the  body  moves  through  the  medium, 
say  with  constant  velocity  v,  it  imparts  this  velocity  to  the  particles 
of  the  medium  it  meets.  The  portion  of  the  medium  so  affected  in  the 
unit  of  time  can  be  regarded  as  a  cylinder  of  cross-section  A  and  length 
V,  and  hence  of  mass  pAv.  To  increase  the  velocity  of  this  mass  from 
0  to  f  in  the  unit  of  time  requires,  by  equation  (5)  of  Art.  171,  a  force 

pAv  ■ V  .  „ 

^—  =  pAi^. 

The  retarding  force  of  the  medium  must  be  equal  and  opposite  to  this 
force  multiplied  by  a  coefficient  k  to  take  into  account  various  disturbing 
influences. 

For  small  velocities,  however,  the  resistance  can  be  assumed  pro- 
portional to  the  velocity,  F  =  kv,  the  coefficient  k  to  be  determined  by 
experiment. 

The  above  consideration  is  only  a  very  rough  approximation.  Thus 
the  particles  of  the  medium  are  not  simply  given  the  velocity  v  in  the 
direction  of  motion;  they  are  partly  pushed  aside  and  move  in  curves 
backwards,  causing  often  whirls  or  eddies  alongside  and  behind  the 
body.  If  the  medium  is  a  gas,  it  is  compressed  in  front,  and  rarefied 
behind  the  body;  indeed,  when  the  velocity  is  great  (greater  than  that 
of  sound  in  the  gas),  a  vacuum  will  be  formed  behind  the  body.  More- 
over, a  layer  of  the  medium  adheres  to  and  moves  with  the  body,  thus 
increasing  the  cross-section.  It  is  therefore  often  found  necessary  to 
assume  a  more  general  expression  for  the  resistance;  and  this  isj  in 
ballistics,  generally  written  in  the  form 

F  =  KpAv'^fiv). 


292. 


MOTION  OF  A  FREE  PARTICLE  223 


The  careful  experiments  that  have  been  made  to  determine  the  re- 
sistance offered  by  the  air  to  the  motion  of  projectiles  have  shown  that 
for  velocities  up  to  about  250  meters  per  second,  as  well  as  for  velocities 
above  420  m./sec,  J{v)  can  be  regarded  as  constant,  i.  e.  the  resistance 
is  proportional  to  the  square  of  the  velocity.  But  for  velocities  between 
250  and  420  m./sec,  i.  e.  in  the  vicinity  of  the  velocity  of  sound  in  air 
(330-340  m./sec),  the  law  of  resistance  is  more  complicated. 

292.  Falling  Body  in  Resisting  Medium.  Assuming  the 
resistance  proportional  to  the  square  of  the  velocity,  the 
equation  of  motion  for  a  body  falling  (without  rotating)  in  a 
medium  of  constant  density  is 

d^s  dv  ,  „ 

at  at 

where  A;  is  a  positive  constant.     To  simplify  the  resulting 
formulae,  put 

g 

then  the  separation  of  the  variables  v  and  t  gives 

gdv 
g^  —  jjL^v^ 
whence 

2m      g  -  tJ-v 

the  constant  of  integration  being  zero  if  the  initial  velocity 
is  zero.     Solving  for  v,  we  have 


V  =  - 


of*'  n 

^^  =  —  tanhjui. 


Writing  dsjdt  for  v  and  integrating  again,  we  find,  since  s  =  0 
for  t  =  0, 

s  =  ~  log  h  (e*^'  +  e"*^')  =    ■,  log  coshyuf. 


224  KINETICS  [293. 

The  relation  between  v  and  s  can  be  obtained  by  eliminating 
t  between  the  expressions  for  v  and  s,  or  more  conveniently 
by  eliminating  /  from  the  original  differential  equation  by 
means  of  the  relation 

dv  _  dvds  _    dv 
dt       ds  dt         ds 
This  gives 

whence,  with  y  =  0  for  s  =  0, 

s  =  -%  log   9         9  2  • 
2m-       g-  -  fi-v^ 

293.  Exercises. 

(1)  Show  that,  as  t  increases,  the  motion  considered  in  Art.  292 
approaches  more  and  more  a  state  of  uniform  motion  without  ever 
reaching  it. 

(2)  Determine  the  motion  of  a  body  projected  vertically  upward  in 
the  air  with  given  initial  velocity  vo,  the  resistance  of  the  air  being  pro- 
portional to  the  square  of  the  velocity. 

(3)  In  iTx.  (2)  find  the  whole  time  of  ascent  and  the  height  reached 
by  the  particle. 

(4)  Show  that,  owing  to  the  resistance  of  the  air,  a  body  projected 
vertically  upward  returns  to  the  starting  point  with  a  velocity  less  than 
the  initial  velocity  of  projection. 

(5)  A  ball,  6  in.  in  diameter,  falls  from  a  height  of  300  ft.;  find 
how  much  its  final  velocity  is  diminished  by  the  resistance  of  the  air,  if 
k  =  0.0C090. 

(6)  Determine  the  rectilinear  motion  of  a  body  in  a  medium  whose 
resistance  is  proportional  to  the  velocity,  when  no  other  forces  act  on  it. 

(7)  A  body  falls  from  rest  in  a  medium  whose  resistance  is  propor- 
tional to  the  velocity;  find  v  and  s  in  terms  of  t,  v  in  terms  of  s. 

294.  Damped  Oscillations.  Let  a  particle  of  mass  m  be 
attracted  by  a  fixed  center  0,  with  a  force  proportional  to 
the  distance  from  0,  and  move  in  a  medium  w^hose  resistance 


294.]  MOTION  OF  A  FREE  PARTICLE  225 

is  proportional  to  the  velocity.  If  the  initial  velocity  be 
directed  through  0  (or  be  zero) ,  the  motion  will  be  rectilinear, 
and  the  equation  of  motion  is 

(P'S 

m-r;;,  =  —  mK}s  —  mkv, 
or,  putting  k  =  2X, 

f  +  2x*;  +  .=.  =  o.  rii) 

This  is  a  homogeneous  linear  differential  equation  of  the 
second  order  with  constant  coefficients,  which  can  be  in- 
tegrated by  a  well-known  process.     The  roots  of  the  auxiliary 

equation,  

-  X±  VX2  -  k\ 

are  real  or  imaginary  according  as  X  >  k,  or  X  <  k.  The 
limiting  cases  X  =  k,  X  =  0,  /c  =  0,  also  deserve  special  men- 
tion. 

(a)  If  X  >  K,  the  roots  are  real  and  different,  and  as  X  is 
positive,  both  roots  are  negative;  denoting  them  by  —  a  and 
—  6,  so  that  a  and  h  are  positive  constants,  and  b  >  a,  the 
general  solution  is 

S   =   CiC""'   +   €26"''^. 

As  the  force  has  a  finite  value  at  the  center  0,  we  can  take 
s  =  0,  V  ^  vo  ior  t  =  0  as  initial  conditions.     This  gives 

s  =  T-^°-  --  (e-°'  -  e-^').     V  =  T^^^^  (he-""  -  ae""'). 
b  —  a  b  —  a 

The  velocity  reduces  to  zero  at  the  time 

h  =  r log-- 

b  —  a       a 

16 


226  KINETICS  (294. 

As  a  and  b  are  positive  and  b  >  a,  s  has  always  the  sign  of 
i'o,  i'  e.  the  particle  remains  always  on  the  same  side  of  0; 
it  reaches  its  elongation  at  the  time  ti,  for  which  v  vanishes, 
and  then  approaches  the  point  0  asymptotically. 

Hence,  in  this  case,  the  damping  effect  of  the  medium  is 
sufficiently  great  to  prevent  actual  oscillations.  Such  mo- 
tions are  sometimes  called  aperiodic. 

(6)  If  X  =  K,  the  roots  are  real  and  equal,  viz.  =  —  X, 
and  the  general  solution  is 

s  =  (ci  +  C2t)e~^. 
With  s  =  0,v  =^  voiov  t  =  0,  we  find 

s  =  wote"^',     V  =  i'o(l  -  X^e-'^'. 
The  velocity  vanishes  for  ti  =  1/X,  and  then  only.     The 
nature  of  the  motion  is  essentially  the  same  as  in  the  previous 
case. 

(c)  If  X  <  K,  the  roots  are  complex,  say  =  —  «  ±  ^i,  where 
a  and  /3  are  positive  constants.     The  general  solution 

s  =  e~"'(ci  coS(8i  +  Co  sinjS^) 
gives  with  s  =  0,  v  =  Vq  for  ^  =  0: 

s  =  ^e-«'  sin/3^,     v  =  -^  e-"'(/3  cos^t  -  a  sin^t). 


Here  v  vanishes  whenever  tan/3^  =  (3 /a  =  V(k/X)^  ~  1;  s 
vanishes  (i.  e.  the  particle  passes  through  0)  whenever  /  is  an 
integral  multiple  of  7r//3;  s  has  an  infinite  number  of  maxima 
and  minima  whose  absolute  values  rapidly  diminish. 

The  resistance  of  the  medium,  while  not  sufficient  to  ex- 
tinguish the  oscillations,  continuall}^  shortens  their  amplitude ; 
this  is  the  typical  case  of  damped  oscillations. 

(d)  If  X  =  0,  the  roots  are  purely  imaginary,  viz.  =  ±  ki. 
In  this  case,  the  second  term  in  equation  (11)  is  zero;  there 


296.]  MOTION   OF  A  FREE   PARTICLE  227 

is  no  damping  effect,  and  we  have  the  case  of  free  oscillations 
(see  Arts.  284-290). 

(e)  If  K  =  0,  one  of  the  roots  is  zero,  the  other  is  =  —  2  X. 
The  attracting  (or  elastic)  force  being  zero,  we  have  the  case 
of  Ex.  (6),  Art.  293. 

295.  As  shown  in  Arts.  273,  274,  the  principle  of  the  conservation 
of  energy  holds  for  the  free  oscillations  of  a  particle  (under  a  force  pro- 
portional to  the  distance).  In  the  case  of  damped  oscillations  (Art.  294), 
this  principle,  in  the  restricted  sense  in  which  it  has  been  proved  so  far, 
is  not  applicable,  the  resistance  of  the  medium  not  being  given  as  a 
function  of  the  distance  s.  The  total  energy  E  =  T  +  V  oi  the  particle, 
or  rather  the  energy  stored  in  the  system  formed  by  the  spring  with 
the  particle  attached  (in  the  example  used  above),  diminishes  in  the 
course  of  time  because  the  spring  has  to  do  work  against  the  resistance 
of  the  medium,  thus  transferring  part  of  its  energy  to  the  medium 
(setting  it  in  motion,  heating  it,  etc.).  Thus,  in  a  generalized  meaning, 
the  principle  of  the  conservation  of  energy  can  be  said  to  hold  for  the 
larger  system,  formed  by  the  spring,  together  with  the  medium  (see 
Art.  276). 

The  rate  at  which  the  total  energy  E  diminishes  with  the  time  is 
here  proportional  to  the  square  of  the  velocity : 

d^  o     ^  2. 

-^  =   -  2?ttXz;2; 

for,  substituting  for  E  its  value  E  =  T  +  V  =  imv^  +  Imk'^s'^   (Art. 
289)  and  reducing,  we  find  the  equation  of  motion  (11). 

The  space-rate  of  change  of  the  total  energy  E  is  proportional  to 
the  velocity,  and  is  nothing  else  but  the  resistance  of  the  medium : 

-r-  =  —  2  m\v, 
ds 

for  we  have 

dE  ^  dE^ds  ^    dE 

dt         ds  dt  ds 

296.  Forced  Oscillations.  In  the  case  of  free  simple  har- 
monic oscillations,  while  the  force  regarded  as  a  function  of 


228  KINETICS  [297. 

the  distance  s  is  directly  proportional  to  s,  the  same  force 
regarded  as  a  function  of  the  time  is  of  the  form 

R  =  —  mK~So  coskI, 

since  s  =  So  cosk^.  Conversely,  a  particle  acted  upon  by  a 
single  force  R  =  mk  cosjit,  or  R  =  mk  shifxt,  directed  toward 
a  fixed  center  0,  will,  if  the  initial  velocity  passes  through  0, 
have  a  simple  harmonic  motion. 

Suppose  that  such  a  force  in  the  line  of  motion  be  super- 
imposed in  the  case  of  Art.  294  so  that  the  equation  of  motion 

becomes 

cPs 
''i  ^7^  =  ~  mii~s  —  2m\v  -\-  mk  cos/jLt, 
dr 

or 

g  +  2X  'J^  +  K^s  =  A;  sin/zf.  (12) 

The  particle  is  then  said  to  be  subject  io  forced  oscillations. 
For  a  particle  suspended  from  a  spiral  spring  this  could  be 
realized  by  subjecting  the  point  of  suspension  to  a  vertical 
simple  harmonic  motion  of  amplitude  k  and  period  27r/ju. 

The  non-homogeneous  linear  differential  equation  (12)  with 
constant  coefficients  can  be  integrated  by  well-known 
methods. 

297.  Exercises. 

(1)  With/x  =  2,  ^0  =  4,  sketch  the  curves  representing  s  as  a  function 
of  t  in  the  five  cases  of  Art.  294;  take  (a)  X  =  3,  (6)  X  =  2,  (c)  \  =  H, 
(e)  X  =  2. 

(2)  Compare  the  cases  (c)  and  (d)  of  Art.  294;  show  that  the  os- 
cillations in  a  resisting  medium  are  isochronous,  but  of  greater  period 
than  in  vacuo.  The  ratio  of  the  amplitude  at  any  time  to  the  initial 
amplitude  is  called  the  dam-ping  ratio;  show  that  the  logarithm  of  this 
ratio,  the  so-called  logarithmic  decrement,  is  proportional  to  the  time. 

(3)  Derive  the  equation  of  motion  in  the  case  of  free  oscillations 
from  the  principle  of  the  conservation  of  energy. 


299.]  MOTION  OF  A  FREE   PARTICLE  229 

(4)  Integrate  and  discuss  the  equation  s  -\-  k^s  =  a  sinjui;  show  that 
the  amphtude  of  the  forced  oscillation  becomes  very  large  if  the  periods 
of  the  free  and  forced  oscillations  are  nearly  equal.  Discuss  the  limiting 
case  when  fx  =  k. 

(5)  Integrate  (12),  assuming  a  particular  integral  of  the  form 
c  cosul  +  c'  siufjil  and  determining  the  constants  c,  c'  by  substituting 
this  expression  in  (12).     Discuss  the  result, 

3.  Examples  of  curvilinear  motion. 

298.  Central  Forces.  The  motion  of  a  particle  in  the  field 
of  a  central  force  has  been  studied  in  Kinematics,  under 
central  motion,  Arts.  96-113.  It  will  here  suffice  to  add 
certain  further  developments  that  are  best  expressed  in 
dynamical  terms. 

299.  Force  Proportional  to  the  Distance :  f(r)  =  /cV.  The  equations 
of  motion  (2)  are  in  this  case 

the  upper  sign  holding  for  attraction,  the  lower  for  repulsion.  Their 
solution  is  very  simple,  because  each  equation  can  be  integrated  sepa- 
rately.    We  find,  in  the  case  of  attraction, 

X  =  Ci  cosK<  +  a-z  sind,     y  =  bi  cosk<  +  62  siuKt, 

and  in  the  case  of  repulsion, 

X  =  aiC'  +  026-"^',     y  =  bie'^'  +  hiC-"*; 

di,  Oi,  61,  hi,  being  the  constants  of  integration. 

To  find  the  equation  of  the  orbit,  it  is  only  necessary  to  eliminate 
t  in  each  case. 

In  the  case  of  attraction,  this  elimination  can  be  performed  by  solving 
for  cosd,  sinx/,  squaring  and  adding.     The  result  is 

{aiy  —  bixy  +  {aiy  —  bnx)"^  =  (aJh  —  aihi)-, 
and  this  represents  an  ellipse,  since 

(fli^  +  a22)(6,2  +  62=)  -  {aA  +  aMY  =  (aA  -  aA)' 
is  always  positive.     The  center  of  the  ellipse  is  at  the  origin,  and  the 
lines  aiy  =  b\x,  aiy  =  box  are  a  pair  of  conjugate  diameters. 


230  KINETICS  1300. 

In  the  case  of  repulsion,  solve  for  c'  and  e-'^',  and  multiply.  The 
resulting  equation, 

(ciy  —  bix)(b2X  —  aoy)  =  (0162  —  a2&i)-, 

represents  a  hj'perbola  whose  asymptotes  are  the  lines  aiy  =  bix, 
a^y  =  62X. 

300.  It  is  worthy  of  notice  that  the  more  general  problem  of  the 
motion  of  a  particle  attracted  by  any  ymmbcr  of  fixed  centers,  icith  forces 
directly  proportional  to  the  distances  from  these  centers,  can  be  reduced 
to  the  problem  of  Art.  299. 

Let  X,  y,  z  be  the  co-ordinates  of  the  particle,  r,  its  distance  from  the 
center  0,-;  x,,  yi,Zi  the  co-ordinates  of  Oi]  and  Krri  the  acceleration 
produced  by  0,.     Then  the  x-component  of  the  resultant  acceleration  is 

=  —  S/v-rr;  .  ^  ~— '  =  -  2K-.2(.r  —  x.)  =  -  x^Kr  +  Sk-.^x;; 

and  similar  expressions  obtain  for  the  y  and  z  components.  Hence, 
the  equations  of  motion  are 

X  =  —  x'S.Kr  +  ^Ki-Xi,    y  =  —  y'^Ki^-  +  'ZK^yi,     z  =  -  z'^kC-  +  'Zki'^zu 

As  the  right-hand  members  are  linear  in  x,  y,  z,  there  is  one,  and  only 
one,  point  at  which  the  resultant  acceleration  is  zero.  Denoting  its 
co-ordinates  by  x,  y,  z,  we  have 

The  form  of  these  equations  shows  that  this  point  of  zero  acceleration 
which  is  sometimes  called  the  mean  center  is  the  centroid  of  the  centers 
of  force,  if  these  centers  be  regarded  as  containing  masses  equal  to  kj^. 
It  is  evidently  a  fixed  point. 

By  introducing  the  co-ordinates  of  the  mean  center,  we  can  reduce 
the  equations  of  motion  to  the  simple  form 

X  =  -k2(x  -  x),     y  =  -  K'^iy  -  y),     2  =  -  K^iz  -  z), 

where  k-  =  Skj-.     Finally,  taking   the  mean  center  as  origin,  we  have 

X  =  —  K^x,     ij  =  —  K-y,     2  =  —  K^z. 

It  thus  appears  that  the  motion  of  the  particle  is  the  same  as  if  there  rvere 
only  a  single  center  of  force,  viz.,  the  mean  center  (x,  y,  z),  attracting  loith 
a  force  proportional  to  the  distance  from  this  center. 


302.]  MOTION   OF   A   FREE   PARTICLE  231 

The  plane  of  the  orbit  is,  of  course,  determined  by  the  mean  center 
and  the  initial  velocity. 

301.  It  is  easy  to  see  that  most  of  the  considerations  of  Art.  300 
apply  even  when  some  or  all  of  the  centers  repel  the  particle  with  forces 
proportional  to  the  distance.  It  may,  however,  happen  in  this  case  that 
the  mean  center  lies  at  infinity,  in  which  case,  of  course,  it  can  not  be 
taken  as  origin. 

Simple  geometrical  considerations  can  also  be  used  to  solve  such 
problems.     Thus,  in  the  case  of  two  attractive  centers  Oi,  Oi  (Fig.  73) 
of  equal  intensity  k^,  the  forces  can 
evidently  be  represented  by  the  dis- 
tances  POi  =  ri,    POi  =  Ti   of    the 
particle  P  from  the  centers.     Their 
resultant  is  therefore  =  2P0,  if  0 
denotes  the  point  midway  between 
OiandO-;  and  this  resultant  always        --n~-- 
passes  through   this  fixed    point  0, 

and  is  proporlional  to  the  distance  -p.     „„ 

PO  from  this  point. 

302.  Exercises. 

(1)  Determine  the  constants  of  integration  in  Art.  299,  if  xo,  2/0  are 
the  co-ordinates  of  the  particle  at  the  time  I.  =  0  and  Vi,  V2  the  com- 
ponents of  its  velocity  Vo  at  the  same  time.  The  equation  of  the  orbit 
will  assume  the  form 

K~{xuy  -  yox)-  +  {ivj  -  VixY  =  (w2  -  rjaihY 
for  attraction,  and 

nKxoy  -  VaxY  -  {viy  -  Vixy  =  -  {xaih  -  yoviy 
for  repulsion. 

(2)  Show  that  the  semi-diameter  conjugate  to  the  initial  radius 
vector  has  the  length  Vu/k,  where  Vu-  =  fi^  -f-  V2^.  As  any  point  of  the 
orbit  can  be  regarded  as  initial  point,  it  follows  that  the  velocity  at  any 
'point  is  proportional  to  the  parallel  diameter  of  the  orbit. 

(3)  Find  what  the  initial  velocity  must  be  to  make  the  orbit  a  circle 
in  the  case  of  attraction,  and  an  equilateral  hyperbola  in  the  case  of 
repulsion. 

(4)  The  initial  radius  vector  ro  and  the  initial  velocity  Vo  being  given 
geometrically,  show  how  to  construct  the  axes  of  the  orbit  described 


232  laNETICS  1302. 

under  the  action  of  a  central  force  (of  given  intensity  k^)  proportional 
to  the  distance  from  the  origin. 

(5)  A  particle  describes  an  ellipse  under  the  action  of  a  central 
force  proportional  to  the  distance;  show  that  the  eccentric  angle  is 
proportional  to  the  time,  and  find  the  corresponding  relation  for  a 
hyperbolic  orbit. 

(6)  A  particle  of  mass  m  describes  a  conic  under  the  action  of  a 
central  force  F  =  =F  mK-r.  Show  that  the  sectorial  velocity  is  ic  = 
},Kah,  a  and  h  being  the  semi-axes  of  the  conic. 

(7)  In  Ex.  (6)  show  that  the  time  of  revolution  is'  T  =  2wIk,  if  the 
conic  is  an  ellipse. 

(8)  A  particle  describes  a  conic  under  the  action  of  a  force  whose 
direction  passes  through  the  center  of  the  conic.  Show  that  the  force 
is  proportional  to  the  distance  from  the  center. 

(9)  A  particle  is  acted  upon  by  two  central  forces  of  the  same 
intensity  (k^),  each  proportional  to  the  distance  from  a  fixed  center. 
Determine  the  orbit:  (a)  when  both  forces  are  attractive;  (6)  when 
both  are  repulsive;  (c)  when  one  is  an  attraction,  the  other  a  repulsion. 

(10)  A  particle  of  mass  m  is  attracted  by  two  centers  0\,  O2  of  equal 
mass  m'  and  repelled  by  a  third  center  O3,  whose  mass  is  m"  =  2m'. 
If  the  forces  are  all  directly  proportional  to  the  respective  distances, 
determine  and  construct  the  orbit. 

(11)  When  a  particle  moves  in  an  ellipse  under  a  force  directed 
towards  the  center,  find  the  time  of  moving  from  the  end  of  the  major 
axis  to  a  point  whose  polar  angle  is  d. 

(12)  Prove  that  if,  in  the  problem  of  Art.  301,  the  intensities  of  Oi  and 
O2  are  ki,  k2,  the  resultant  attraction  F  passes  through  the  centroid  G 
of  two  masses  ki  k2,  placed  at  Oi,  O2,  and  that  F  =  {ki  +  k2)PG. 

(13)  In  Art.  299,  in  the  case  of  attraction,  the  component  motions  are 
evidently  simple  harmonic  oscillations.  Show  that  the  equation  of  the 
path  can  be  put  in  the  form  (comp.  Art.  89) 

x^      2xy  .  ^   ,  y^ 

— , r  sm5  +  r,  =  cos^S. 

a'       ab  W 

(14)  Show  that  the  total  energy  of  a  particle  of  mass  m  describing  an 
ellipse  of  semi-axes  a,  b  under  a  force  ninh  directed  to  the  center  is 


304.]  MOTION  OF  A  FREE  PARTICLE  233 

303.  Force  Inversely  Proportional  to  the  Square  of  the 
Distance :  /(r)  =  ^/j"'  (Newton's  law). 

It  has  been  shown  in  Kinematics  (Arts.  99-108)  how  this 
law  of  acceleration  can  be  deduced  from  Kepler's  laws  of 
planetary  motion.  From  Kepler's  first  law  Newton  con- 
cluded that  the  acceleration  of  a  planet  (regarded  as  a  point 
of  mass  7n)  is  constantly  directed  towards  the  sun;  from 
the  second  he  found  that  this  acceleration  is  inversely  pro- 
portional to  the  square  of  the  distance.  The  motion  of  a 
planet  can  therefore  be  explained  on  the  hypothesis  of  an 
attractive  force, 

^  =  ^^^2* 
issuing  from  the  sun 

The  value  of  n,  which  represents  the  acceleration  at  unit 
distance  or  the  so-called  intensity  of  the  force,  was  found  to 
be  (Art.  108;  or  below,  Art.  315) 


M  =  4x2—; 

and  as,  according  to  Kepler's  third  law,  the  quantity  a^/T"^ 
has  the  same  value  for  all  the  planets,  Newton  inferred  that 
the  intensity  of  the  attracting  force  is  the  same  for  all 
planets;  in  other  words,  that  it  is  one  and  the  same  central 
force  that  keeps  the  different  planets  in  their  orbits. 

304.  It  was  further  shown  by  Newton  and  Halley  that  the 
motions  of  the  comets  are  due  to  the  same  attractive  force. 
The  orbits  of  the  comets  are  generally  ellipses  of  great  eccen- 
tricity, with  the  sun  at  one  of  the  foci.  As  a  comet  is  within 
range  of  observation  only  while  in  that  portion  of  its  path 
which  lies  nearest  to  the  sun,  a  portion  of  a  parabola,  with  the 
same  focus  and  vertex,  can  be  substituted  for  this  portion  of 
the  elliptic  orbit,  as  a  first  approximation. 


234  KINETICS  [305. 

It  is  also  found  from  observation  that  the  motions  of  the 
moons  or  satelhtes  around  the  planets  follow  very  nearly 
Kepler's  laws.  A  planet  can  therefore  be  regarded  as  at- 
tracting each  of  its  satellites  with  a  force  proportional  to  the 
mass  of  the  satellite  and  inversely  proportional  to  the  square 
of  the  distance. 

305.  All  these  facts  led  Newton  to  suspect  that  the  force  of 
terrestrial  gravitation,  as  observed  in  the  case  of  falling  bodies 
on  the  earth's  surface,  might  be  the  same  as  the  force  that 
keeps  the  moon  in  its  orbit  around  the  earth.  This  inference 
could  easily  be  tested,  since  the  acceleration  g  of  falling  bodies 
as  well  as  the  moon's  distance  and  time  of  revolution  were 
known. 

Let  m  be  the  mass  of  the  moon,  a  the  major  semi-axis  of  its  orbit,  T 
the  time  of  revolution,  r  the  tUstance  between  the  centers  of  earth  and 
moon;  then  the  earth's  attraction  on  the  moon  is  (Art.  303) 

a' 
F  =  Airhn  ^—  , 

or,  since  the  eccentricity  of  the  moon's  orbit  is  so  small  that  the  orbit 

can  be  regarded  as  nearly  circular,  F  =  Air-ma/T-.     On  the  other  hand, 

the  attraction  exerted  by  the  earth  on  a  mass  m  on   its  surface,  i.  e. 

at  the  distance  R  =  3963  miles  from  the  center,  is  F'  =  mg.     Now, 

if  these  forces  are  actually  in  the  inverse  ratio  of  the  squares  of  the 

distances,  we  must  have 

F'  ^  a? 

F   ~  I^' 

or,  since  the  distance  of  the  moon  is  nearly  =  QOR,  F'  =  GO^F.  Sub- 
stituting the  above  values  of  F  and  F',  we  find 

A  2  60^-R 

With  R  =  3963  miles,  T  =  27''  7"  43'",  this  gives  g  =  32.0,  a  value 
which  agrees  sufficiently  with  the  observed  value  of  g,  considering  the 
rough  degree  of  approximation  used. 


308.1  MOTION  OF  A  FREE  PARTICLE  235 

306.  In  this  way  Newton  was  finally  led  to  his  law  of 
universal  gravitation,  which  asserts  that  every  particle  of  mass 
m  attracts  every  other  particle  of  mass  m'  with  a  force 

„         mm' 

where  r  is  the  distance  of  the  particles  and  k  a  constant,  viz. 
the  acceleration  produced  by  a  unit  of  mass  in  a  unit  of  mass 
at  unit  distance  (see  Arts.  245,  246). 

The  best  test  of  this  hypothesis  as  an  actual  law  of  physical 
nature  is  found  in  the  close  agreement  of  the  results  of 
theoretical  astronomy  based  on  this  law  with  the  observed 
celestial  phenomena. 

307.  Taking  Newton's  law  as  a  basis,  let  us  now  turn  to 
the  converse  prol)lem  of  determining  the  7notion  of  a  particle 
acted  7ipon  by  a  single  central  force  for  which  f{r)  =  fi/r'^ 
(problem  of  planetary  motion). 

It  has  been  shown  in  Kinematics  (Arts.  109-112)  that  if 
the  force  be  attractive,  the  particle  will  describe  a  conic  section 
with  one  of  the  foci  at  the  center  of  force,  the  conic  being  an 
ellipse,  parabola,  or  hyperbola,  according  as 

Vo'  =  ^.  (13) 

^  To 

If  the  force  be  repulsive,  the  same  reasoning  will  apply, 
except  that  fj,  is  then  a  negative  quantity.  The  orbit  is, 
therefore,  in  this  case  always  hyperbolic;  the  branch  of  the 
hyperbola  that  forms  the  orbit  must  evidently  turn  its  convex 
side  towards  the  focus  at  which  the  center  of  force  is  situated, 
since  the  force  always  lies  on  the  concave  side  of  the  path. 

308.  To  exhibit  fully  the  determination  of  tlie  constants  and  the 
dependence  of  the  nature  of  the  orbit  on  the  initial  conditions,  a  solution 
somewhat  different  from  that  given  in  Kinematics  will  here  be  given  for 
the  problem  of  planetary  motion  in  its  simplest  form. 


236  KINETICS  (309. 

With  /(r)  =  M/r^,   the  equation  of  kinetic   energy   and   work    (5) 
Art.  271,  gives  (comp.  (19),  Art.  109) 


'      r^  r         To 


or,  if  the  constant  of  integration  be  denoted  briefly  by  h  and  u  =  1/r  be 
introduced, 

1^2  =  2u,u  +  /i,   where  /i  =  v^? •  (14) 

Substituting  this  expression  for  v-  in  the  equation  (15),  Art.  105,  we 
find  the  differential  equation  of  the  orbit  in  the  form 

^^^y  +  u^  =  \,{2y.u  +  h),  (15) 


( 


(^)'-(»-;)'+';:+' 


To  integrate,  we  introduce  a  new  variable  u'  by  putting 


u  ,     1 1^^       h 

the  resulting  equation, 

(  '^  )  =  1  _  ,i'2  or  dB  =±     , . 

\  dd  J  >/ 1  -  u'2 

has  the  general  integral 

0  —  a  =  =F  cos-' a',   or  u'  =  cos (9  —  a), 

where  a  is  the  constant  of  integration.  The  orbit  has,  therefore,  the 
equation 

^    =-.  +  ^!-,+^cos{8-a),  (16) 

r        c-        y  c*       c^ 

which  agrees  with  the  equation  (24)  given  in  Kinematics,  Art.  112, 
excepting  the  different  notation  used  for  the  constants. 

309.  The  equation  (16)  represents  a  conic  section  referred  to  its 
focus  as  origin.     The  general  focal  equation  of  a  conic  is 

~  =\+l  cos(0  -  a),  (17) 

where  I  is  the  semi-latus  rectum,  or  parameter,  e  the  eccentricity,  and 
a  the  angle  made  with  the  polar  axis  by  the  line  joining  the  focus  to  the 
nearest  vertex. 


310.1 


MOTION   OF   A   FREE   PARTICLE 


237 


In  a  planetary  orbit  (Fig.  74),  the  sun  S  being  at  one  of  the  foci,  the 
nearest  vertex  A  is  called  the  perihelion,  the  other  vertex  A'  the  aphelion, 
and  the  angle  d  —  a  made  by  any  radius  vector  SP  =  r  with  the  peri- 
helion distance  SA  is  called  the  true  anomaly. 


Fig.  74. 


Comparing  equations  (17)  and  (IG),  we  find,  for  the  determination  of 
the  constants: 

I        c2  '      I        ^  c*'^  c^  ' 
hence, 

l^'\    e=Jl+^,  (18) 

or,  solving  for  c  and  h, 

c2  -  1 


C  =   'SiJd,     h  =  IX 


I 


(19) 


310.  The  expression  for  the  eccentricity  e  in  (IS)  determines  fhe 
nature  of  the  conic;  the  orbit  is  an  ellipse,  parabola,  or  hyperbola, 
according  as  e  =  1;  hence,  by  (18),  according  as  the  constant  h  of  the 
equation  of  kinetic  energy  is  negative,  zero,  or  positive.  Owing  to  the 
value  of  h  given  in  (14),  this  criterion  agrees  with  the  form  (13),  Art. 
307. 

It  should  be  observed  that  it  follows  from  (13)  that  the  nature  of  the 
conic  is  independent  of  the  direction  of  the  initial  velocity. 

The  criterion  (13)  can  be  given  the  following  interpretation.     Con- 


238  KINETICS  [311. 

sider  a  particle  attracted  by  a  fixed  center  according  to  Newton's 
law.  If  it  move  in  a  straight  line  passing  through  the  center,  the 
principle  of  kinetic  energy  gives  for  its  velocity,  at  the  distance  r, 


-  2.  r-^  - 


V-  =  vo"^  —  2{i.  \     '^  =  ^  -{-  vo'  — 


hence,  if  it  start  from  rest  at  an  infinite  distance  from  the  center,  it 
would  acquire  the  velocity  V  2iJ.ir  at  the  distance  r.  The  criterion  (13) 
is  therefore  equivalent  to  saying  that  the  orbit  is  an  ellipse,  a  parabola, 
or  a  hyperbola,  according  as  the  velocity  at  any  point  is  less  than,  equal 
to,  or  greater  than  the  velocity  which  the  particle  would  have  acquired  at 
that  point  by  falling  towards  the  center  from  infinity. 

311.  For  a  central  conic,  whose  axes  are  2a,  2b,  we  have  I  =  b-Ja. 
e  =  V  a^  =F  ¥/a  (the  upper  sign  relating  to  the  ellipse,  the  lower  to  the 
hyperbola),  so  that  the  equations  (19)  reduce  to  the  following: 

c=6>,     h  =  =f^.  (20) 

V  a  a 

The  latter  relation,  with  the  value  of  h  from  (14),  gives  for  the  major 
or  focal  semi-axis  a : 

±^=?--^^;  (21) 

a       ro        fi 

while  the  former,  with  the  value  of  c  as  given  in  Art.  100,  determines 
the  minor  or  transverse  axis  b : 

b  =  c\j       =  ?v'o  sini/'o  \/     •  (22) 


a/       =  rovo  siniAo  \/  ' 


312.  The  magnitudes  of  the  axes  having  thus  been  found,  their 
directions  can  be  determined  by  a  simple  construction  which  furnishes 
the  second  focus. 

In  the  ellipse,  the  focal  radii  have  a  constant  sum  =  2a,  and  lie  on 
the  same  side  of  the  tangent,  making  equal  angles  with  it.  In  the 
hyperbola,  they  have  a  constant  difference  =  2a,  and  lie  on  opposite 
sides  of  the  tangent. 

Hence,  determining  the  point  0"  (Fig.  75),  which  is  symmetrical 
to  the  center  of  force  O  with  respect  to  the  initial  velocity,  and  drawing 
the  line  PoO",  we  have  only  to  lay  off  on  this  line  from  Po  a  length 
PoO'  =  ±  {2a  —  ro) ;    then  0'  is  the  second  focus,  which  for  an  elliptic 


314.] 


MOTION   OF   A  FREE   PARTICLE 


239 


orbit  must  be  taken  with  O  on  the  same  side  of  the  tangent  PoT",  and  for 
a  hyperbolic  orbit  on  the  opposite  side. 


Fig.  75. 


313.  For  a  parabola,  since  c  =  1,  we  find,  from  (19), 
,        „    ,       c^         Vo~ro^  sinVo 


(23) 


The  axis  of  the  parabola  is  redtlily  found  by  remembering  that  the 
perpendicular  let  fall  from  the  focus  on  the  tangent  bisects  the  tangent 
(i.  e.  the  segment  of  the  tangent  between  the  point  of  contact  and  the 
axis).  Hence,  if  Or  (Fig.  76)  be  the 
perpendicular  let  fall  from  the  center 
0  on  the  velocity  vo,  it  is  only  neces- 
sary to  make  TT'  =  P^T,  and  T'  will 
be  a  point  of  the  axis.  Moreover,  the 
perpendicular  let  fall  from  T  on  OT'  will 
meet  the  axis  at  the  vertex  A  of  the  para- 
bola, so  that  OA  =  ^l. 

314.  The  relation  (21),  which  must 
evidently  hold  at  any  point  of  the  or- 
bit, can  be  written  in  the  form 


Fig.  76. 


■^^Oh)' 


(24) 


the  upper  sign  relating  to  the  ellipse,  the  lower  to  the  hyperbola,  while 
for  the  parabola,  the  second  term  in  the  parenthesis  vanishes  (since 
a  =  oo). 

This  convenient  expression  for  the  velocity  in  terms  of  the  radius 
vector  might  have  been  derived  directly  from  the  fundamental  relation 
(Art.  100)  V  =  c/p,  the  first  of  the  equations  (19),  c^  =  fil,  and  the 


240  KINETICS  1315. 

geometrical  properties  of  the  conic  sections  {r  :^r'  =  2a,  pp'  =  ¥, 
p'r  =  pr',  where  r,  r'  are  the  focal  radii,  and  p,  p'  the  perpendiculars 
let  fall  from  the  foci  on  the  tangent).  The  proof  is  left  to  the  student. 
315.  Time.  In  the  case  of  an  elliptic  orbit,  the  time  7"  of  a  complete 
revolution,  usually  called  the  periodic  time,  is  found  by  remembering 
that  the  sectorial  velocity  is  constant  and  =  ^c,  whence 

_  2Trab 
c    ' 
or,  by  (20), 

T  =  2xV'"'=?^.  (25) 

\  /J,        n 


The  constant  _ 


^  =  \, 


which  evidently  represents  the  mean  angular  velocity  about  the  center 
in  one  revolution,  is  called  the  mean  motion  of  the  planet.  It  should 
be  noticed  that  it  depends  not  only  on  the  intensity  of  the  force,  but 
also  on  the  major  axis  of  the  orbit,  while  in  the  case  of  a  force  directly 
proportional  to  the  distance  the  periodic  time  is  independent  of  the 
size  of  the  orbit  (see  Art.  302,  Ex.  7). 

The  periodic  time  T  and  the  major  axis  a  of  a  planetary  orbit  deter- 
mine the  intensity  n  of  the  force: 

M=47r^^3,  (26) 

whence 

F  =  mJ{r)=mt^  =  Ai^^m^^,  (27) 

where  m  is  the  mass  of  the  planet. 

316.  To  find  generally  the  time  t  in  terms  of  B  or  r,  it  is  best  to  intro- 
duce the  eccentric  angle  ^  of  the  elUpse  as  a  new  variable,  and  to  express 
/,  r,  and  6  in  terms  of  <j>.  In  astronomy,  the  polar  angle  6  is  known  as 
the  true  anomaly,  and  the  eccentric  angle  <j>  as  the  eccentric  anomaly. 

The  relation  of  the  eccentric  angle  ^  to  the  polar  co-ordinates  r,  0 
will  appear  from  Fig.  77,  in  which  P  is  the  position  of  the  planet  at  the 
time  t,  P'  the  corresponding  point  on  the  circumscribed  circle,  i^AOP  = 
6  the  true  anomaly,  and  ^ACP'  =  </>  the  eccentric  anomaly.  The 
focal  equation  of  the  ellipse 

^  I  ^  ail  -  e2) 

1  +  e  cos9      1  +  e  COS0 


317.] 


MOTION   OF  A   FREE   PARTICLE 


241 


gives  r  -{■  er  cos^  =  a  —  ae-;  and  the  figure  shows  that  r  cos0  =  a  cos</) 

—  ae;  hence 

r  =  a(l  —  e  COS0),     or     a  —  r  =  ae  cos(p.  (28) 

Equating  this  value  of  r  to  that  given  by  the  polar  equation  of  the 
ellipse,  we  have 


1  —  e  cos<P 


1  -e2 


or     COS0  = 


cos<^  —  e 


1  +  e  cos9'     "'         ""        I  —  e  COS0  ' 
A  more  symmetrical  form  can  be  given  to  this  relation  by  computing 

1  —  cosO  =  2  sm-^0  =  (1  +  e)  .— 


1  —  e  cos^  ' 


1  +  C089  =  2  cos2^0  =:(1  -  e)  ,^  ^  ^"^"^  ; 

1  —  e  cos^ 


whence,  by  division, 


tanie  =  -v/- tani0. 

\  1  —  e 


(29) 


317.  To  find  t  in  terms  of  r,  we  have  only  to  substitute  in  (24)  for 
v^  its  value  (Art.  105),  and  to  integrate  the  resulting  differential  equation 


Fig.  77. 


(I) 


+ 


r-        r        a 
As,  by  (20),  Art.  311,  c^  =  iJ.b~/a  =  ^0(1  —  c^),  this  equation  becomes 


Kt)'=:i°"'-(«-^"' 


d< 


rdr 


A/x  /aV  —  (a  —  r)2' 


17 


242  KINETICS  [318. 

The  integration  is  easily  performed  by  introducing  the  eccentric 
angle  <t>  as  variable  by  means  of  (28) ;  this  gives 

dl  =  \     ■■  a(l  —  c  cos<^)d<p. 

If  the  time  be  counted  from  the  perihelion  passage  of  the  planet,  we 
have  t  =  0  when  r  =  a  —  ae,  i.  e.  when  0  =  0;  hence,  putting  y  n/a^ 
=  H,  as  in  Art.  315,  we  find 

nt  =  4>  —  e  sin<^.  (30) 

This  relation  is  known  as  Kepler's  equation;  the  quantity  nt  is  called 
the  mean  anomaly. 

318.  Kepler's  equation  (30)  can  be  derived  directly  by  considering 
that  the  ellipse  APA'  (Fig.  77)  can  be  regarded  as  the  projection  of 
the  circle  AP'A',  after  turning  this  circle  about  AA'  through  an  angle 
=  cos~^  (b/a).  For  it  follows  that  the  elliptic  sector  AOP  is  to  the 
circular  sector  AOP'  as  b  is  to  a.     Now,  for  the  circular  sector  we  have 

AOP'  =  ACP'  -  OCP'  =  W-<j,  -  lae  ■  a  sin<^  =  ^0^(0  -  e  sin0); 

hence,  the  elliptic  sector  described  in  the  time  t  is 

AOP  =  -  ■  AOP'  =   lab  (0  -  e  sin0). 
a 

The  sectorial  velocity  being  constant  by  Kepler's  first  law,  we  have 

AOP  ^TTob. 
1  T' 

hence, 

T 

t  ^    -(<j>  -  e  sm0), 

and  this  agrees  with  (30)  since,  by  (25),  2Tr/T  =  n. 

319.  Kepler's  equation  (30)  gives  the  time  as  a  function  of  0;  by 
means  of  (28),  it  establishes  the  relation  between  t  and  r;  b}'  means 
of  (29),  it  connects  t  with  0.  It  is,  however,  a  transcendental  equation 
and  cannot  be  solved  for  0  in  a  finite  form. 

For  orbits  with  a  small  eccentricity  e,  an  approximate  solution  can 
be  obtained  by  writing  the  equation  in  the  form 

4>  =  nt  -\-  e   sin</), 
and  substituting  under  the  sine  for  4>  its  approximate  value  nt'. 


320.]  MOTION   OF   A   FREE   PARTICLE  243 

4>=^  nt  -{-  e  sinnl.  (31) 

This  amounts  to  neglecting  terms  containing  powers  of  e  above  the 
first  power. 

Substituting  this  vakie  of  4>  in  (28),  we  have  with  the  same  approxi- 
mation 

r  =  a{l  —  e  cosnt).  (32) 

To  find  d  in  terms  of  t,  we  have  from  the  equation  of  the  eHipse, 
r  =  a(l  —  t-)(l  4-  e  cos0)~i  =  a(l  —  e  cos9),  neglecting  again  terms  in 
e^;  hence,  r'^  =0^(1  —  2e  cos9).  Substituting  this  value  in  the  equa- 
tion of  areas,  r^dd  =  cdt  =  V fj.a{l  —  e'^)dl,  we  find 

(1  -  2e  cosd)dd  =  J-^  dl  =  ndt) 

whence,  by  integration,  since  0  =  0  for  i  =  0, 

d  —  2e  sin&  =  nt, 
or  finally, 

d  =  nt  +  2c  shmt.  (33) 

Thus  we  have  in  (31),  (32),  (33)  approximate  expressions  for  (j), 
r,  and  9  directly  in  terms  of  the  time.  The  quantity  2e  sinnt,  by  which 
the  true  anomaly  d  exceeds  the  mean  anomaly  nt,  is  called  the  equation 
of  the  center. 

320.  Exercises. 

(1)  A  particle  is  attracted  by  a  fixed  center  according  to  Newton's 
law.     What  must  be  the  initial  velocity  if  the  orbit  is  to  be  circular? 

(2)  A  number  of  particles  are  projected,  from  the  same  point  in 
the  field  of  a  force  following  Newton's  law,  with  the  same  velocity,  but 
in  different  directions.  Show  that  the  periodic  times  are  the  same  for 
all  the  particles. 

(3)  The  mean  distance  of  Mars  from  the  sun  being  1.5237  times 
that  of  the  earth,  what  is  the  time  of  revolution  of  Mars  about  the  sun? 

(4)  A  particle  describes  a  conic  under  the  action  of  a  central  force 
following  Newton's  law;  if  the  intensity  ^  of  the  force  be  suddenly 
changed  to  ju',  what  is  the  effect  on  the  orbit? 

(.'i)  In  Ex.  (4),  if  the  original  orbit  was  a  parabola  and  the  intensity 
be  doubled,  what  is  the  new  orbit? 

(G)  Regarding  the  moon's  orbit  about  the  earth  as  circular,  what 
would  it  become:  (a)  if  the  earth's  mass  were  suddenly  doubled?  (h)  if 
it  were  reduced  to  one  half? 


244  KINETICS  •  [321. 

(7)  In  Ex.  (4),  determine  the  effect  on  the  major  semi-axis  (or  "mean 
distance")  a  and  on  the  periodic  time  T,  of  a  small  change  in  the 
intensity  m  of  the  force. 

(8)  If  the  mass  M  of  the  sun  be  suddenly  increased  by  Mjn,  n  being 
very  large,  while  the  earth  is  at  the  end  of  the  minor  axis  of  its  orbit, 
what  would  be  the  effect  on  the  earth's  mean  distance  and  on  the 
period   of   revolution    T  ? 

(9)  Find  the  equation  of  the  hodograph  of  planetary  motion, 
derive  from  it  the  expression  for  the  velocity  in  terms  of  the  radius 
vector,  and  show  that  the  velocity  is  a  maximum  in  perihelion  and  a 
minimum  in  aphelion. 

(10)  Show  that  the  greatest  velocity  of  a  planet  in  its  orbit  about  the 
sun  is  to  its  least  velocity  as  1  +  e  is  to  1  —  e;  and  find  this  ratio  for 
the  earth,  whose  orbit  has  the  eccentricity  e  =  0.016  771  2. 

(11)  Find  the  time  exactly  as  a  function  of  0,  for  a  parabolic  orbit. 

(12)  The  latus  rectum  passing  through  the  sun  divides  the  earth's 
orbit  into  two  different  parts;  in  what  time  are  these  described  if  the 
whole  time  is  365  K  days? 

(13)  Show  that  the  path  of  a  projectile  in  vacuo  is  an  ellipse,  parabola, 
or  hyperbola,  according  as  Vts  =  36,800  ft.  per  second  (  =  7  miles 
per  second,  nearly).  One  of  the  foci  lies  at  the  center  of  the  earth, 
and  the  ordinary  assumption  that  the  path  is  parabolic  means  that  this 
center  can  be  regarded  as  infinitely  distant.  Show  also  that  the  path 
becomes  circular  for  v^  =  5  miles  per  second,  nearly. 

321.  The  Problem  of  Two  Bodies.  In  the  preceding  discussion  of 
the  motion  of  a  particle  under  the  action  of  a  central  force,  it  has  been 
assumed  that  the  center  of  force  is  fixed.  In  the  applications  of  the 
theory  of  central  forces  this  assumption  is  in  general  not  satisfied. 
Thus,  in  considering  the  motion  of  a  planet  around  the  sun,  the  force 
of  attraction  is,  according  to  Newton's  law  of  universal  gravitation  (Art. 
306),  regarded  as  due  to  the  presence  of  a  mass  M  at  the  center  (sun), 
and  of  a  mass  m  at  the  attracted  point  (planet);  and  the  action  between 
these  two  masses  is  a  mutual  action,  being  of  the  nature  of  a  stress,  i.  e. 
consisting  of  two  equal  and  opposite  forces,  each  equal  to 

„         mM 
F  =  K  . 

Hence,  the  mass  m  of  the  planet  attracts  the  mass  M  of  the  sun  with 


323.]  MOTION  OF  A  FREE  PARTICLE  245 

precisely  the  same  force  with  which  the  mass  M  of  the  sun  attracts  the 
mass  m  of  the  planet.  The  attraction  affects,  therefore,  the  motions  of 
both  bodies. 

322.  The  accelerations  produced  by  the  two  forces  are,  of  course, 
not  equal.  Indeed,  the  acceleration  F/m  =  kM/t'',  produced  in  the 
planet  by  the  sun,  is  very  much  greater  than  the  acceleration  F/M 
=  Kin/r'^,  produced  by  the  planet  in  the  sun;  for  the  mass  of  even  the 

largest  planet  (Jupiter)  is  less  than  one  thousandth  of  that  of  the  sun. 
The  assumption  of  a  fixed  center  can  therefore  be  regarded  as  a  first 
approximation  in  the  problem  of  the  motion  of  a  planet  about  the  sun. 

In  the  case  of  the  earth  and  moon,  the  difference  of  the  masses  is 
not  so  great,  the  mass  of  the  moon  being  nearly  one  eightieth  of  that 
of  the  earth. 

It  can  be  shown,  however,  that  the  results  deduced  on  the  assumption 
of  a  fixed  center  can,  by  a  simple  modification,  be  made  available  for 
the  solution  of  the  general  -problem  of  the  motions  of  two  particles  of 
masses  m.,  M,  subject  to  no  forces  besides  their  mutual  attraction.  In 
astronomy,  this  is  called  the  problem  of  two  bodies.  In  the  solution 
below  we  assume  the  attraction  to  follow  Newton's  law  of  the  inverse 
square  of  the  distance.  It  will  be  convenient  to  speak  of  the  two 
particles,  or  bodies,  as  planet  (m)  and  sun  (ilf). 

323.  With  regard  to  any  fixed  system  of  rectangular  axes,  let  x,  y,  z 
be  the  co-ordinates  of  the  planet  (m),  at  the  time  t;  x',  y',  z'  those  of 
the  sun  {M),  at  the  same  time;  so  that  for  their  distance  r  we  have 

r2  =  (x  -  x'Y  +  {y  -  y'Y  +  (2  -  z')\ 

Then  the  equations  of  motion  of  the  planet  are 

mx  =  F  •^-^— ,     mij  =  F-^-—-,     mz  =  F  -^    ~  -  ,         (1) 

while  the  equations  of  motion  of  the  sun  are 

Mx'  =  F  .  ^^^^-  ,     Mij'  =  F    y~-y' ,     Mz'  =  F  ■  ^^—  .      (2) 

By  adding  the  corresponding  equations  of  the  two  sets,   we  find 

d^  d-  (/'■* 

^^2  (mx  +  Mx')  =  0,     ^^,  {my  +  My')  =  0,      ^^piz  +  Mz')  =  0. 

If  it  be  remembered  that  the  centroid  of  the  two  masses  m,  M  has  the 
co-ordinates 


246  KINETICS  [324. 

_  _  mx  +  Mx'       ^  _niy  +  My'      _  _  mz  +  Mz' 

it  appears  that  these  equations  can  be  written  in  the  form 
^  =  0      -^  =  0      -  =  0- 

dt^      '    dfi      '   dp      ' 

in  words:  the  acccleralion  of  the  common  centroid  of  planet  and  sun  is 
zero;  i.  e.  this  centroid  moves  with  constant  velocity  in  a  straight  line. 
324.  The  integration  of  the  equations  (1)  would  give  the  absolute 
path  of  the  planet.  But  the  constants  could  not  be  determined,  because 
the  absolute  initial  position  and  velocity  of  the  planet  are,  of  course,  not 
known.  The  same  holds  for  the  absolute  path  of  the  sun.  All  we  can 
do  is  to  determine  the  relative  motion,  and  we  proceed  to  find  the 
motion  of  the  planet  relative  to  the  sun. 

Taking  the  sun's  center  as  new  origin  for  parallel  axes,  we  have  for 
the  co-ordinates  f,  i?,  i'  of  the  planet  in  this  new  system, 

^  =  X  -  x',     V  =  y  -  y',     t  =  z  —  z'. 

Now,  dividing  the  equations  (1)  by  m,  the  equations  (2)  by  Rf,  and  sub- 
tracting the  equations  of  set  (2)  from  the  corresponding  equations  of 
set  (1),  we  find  for  the  relative  acceleration  of  the  planet 

V            M  +  m   ^                     M  +  m   V       -.              il/  +  w    f        ,„, 
k  =  —  K ,     V  =  ~  K  , ,     s'  =  -  K ^ —  •  -" .      (3) 

The  form  of  these  equations  shows  that  the  relative  motion  of  the  planet 
with  respect  to  the  sun  is  the  same  as  if  the  sun  ivere  fixed  and  contained 
the  7nass  M  -\-  m.  Thus  the  problem  is  reduced  to  that  of  a  fixed  center, 
the  only  modification  being  that  the  mass  of  the  center  M  should  be 
increased  by  that  of  the  attracted  particle  ?n. 

325.  This  result  can  also  be  obtained  by  the  following  simple  con- 
.sideration.  The  relative  motion  of  the  planet  with  respect  to  the  sun 
would  obviously  not  be  altered  if  geometrically  equal  accelerations  were 
applied  to  both.  Let  us,  therefore,  subject  each  body  to  an  additional 
acceleration  equal  and  opposite  to  the  actual  acceleration  of  the  sun 
(whose  components  are  obtained  by  dividing  the  equations  (2)  by  M). 
Then  the  sun  will  be  reduced  to  equilibrium,  while  the  resulting  accel- 
eration of  the  planet,  which  is  its  relative  acceleration  with  respect  to 
the  sun,  will  evidently  be  the  sum  of  the  acceleration  exerted  on  it  by 


327.]  MOTION  OF  A  FREE  PARTICLE  247 

the  sun  and  the  acceleration  exerted  on  the  sun  by  the  planet.     This 
is  just  the  result  expressed  by  the  equations  (3). 

326.  It  can  here  only  be  mentioned  in  passing  that,  while  the  problem 
of  two  bodies  thus  leads  to  equations  that  can  easily  be  integrated, 
the  problem  of  three  bodies  is  one  of  exceeding  difficulty,  and  has  been 
solved  only  in  a  few  very  special  cases.  Much  less  has  it  been  possible 
to  integrate  the  3  n  equations  of  the  problem  of  n  bodies. 

327.  According  to  the  equations  (3),  the  first  and  second  laws  of 
Kepler  can  be  said  to  hold  for  the  relative  motion  of  a  planet  about  the 
sun  (or  of  a  satellite  about  its  primary).  The  third  law  of  Kepler 
requires  some  modification,  since  the  intensity  of  the  center  ^  should 
not  be  kM,  but  k{M  +  m).     We  have,  by  (26),  Art.  315, 

M=  k{M  +  ?n)  =  47r2,^; 

in  other  words,  the  quotient  a'/T^  is  not  independent  of  the  mass  m 
of  the  planet. 

Thus,  if  mi,  ra-i  be  the  masses  of  two  planets,  Oi,  ao  the  major  semi- 
axes  of  their  orbits,  and  Ti,  T^  their  periodic  times,  we  have 

OiVTii  _  M  +  m,  ^  1  +  mi/M 

This  quotient  is  approximately  equal  to  1  if  M  is  very  large  in  com- 
parison with  both  nil  and  /«2;  hence,  for  the  orbits  of  the  planets 
about  the  sun,  Kepler's  third  law  is  very  nearly  true. 


CHAPTER  XIV. 
CONSTRAINED   MOTION  OF  A   PARTICLE.J 

1.  Introduction. 

328.  A  free  particle  is  said  to  have  three  degrees  of  freedom 
(Art.  231)  since  three  co-ordinates  are  required  to  determine 
its  position,  and  each  of  these  co-ordinates  can  vary  inde- 
pendently of  the  other  two. 

If  the  co-ordinates  of  a  moving  particle  are  subjected  to 
one  condition,  say 

<p{^,  y,  ^)  =  0,  (1) 

the  particle  is  said  to  have  one  constraint  and  only  two  de- 
grees of  freedom.     It  can  then  only  move  on  the  surface  (1), 
and  its  position  on  this  surface  can  be  assigned  by  two  co- 
ordinates (such  as  latitude  and  longitude  on  a  sphere). 
If  the  co-ordinates  are  subjected  to  two  conditions,  say 

ifix,  7j,  z)  =  0,     \P(x,  y,  z)  =  0,  (2) 

the  particle  has  two  constraints  and  but  one  degree  of 
freedom.  It  can  only  move  along  the  curve  of  intersection 
of  the  two  surfaces  (2),  and  its  position  on  this  curve  can  be 
assigned  by  a  single  co-ordinate  (such  as  the  arc  of  the  curve). 

Three  such  conditions  would  in  general  prevent  the  particle 
entirely  from  moving. 

The  surface  or  curve  to  which  a  particle  is  constrained 
may  vary  its  position  or  even  its  shape  in  the  course  of  the 
motion.  The  equations  (1)  and  (2)  would  then  contain  t  as 
a  fourth  independent  variable.  We  shall,  however,  in  general 
assume  that  the  surface  or  curve  is  fixed. 

248 


330.1         CONSTRAINED   MOTION  OF  A  PARTICLE  249 

329.  A  particle  constrained  to  a  surface  can  be  regarded  as 
the  limit  of  a  small  piece  of  matter  confined  between  two. 
very  near  impenetrable  surfaces.  The  constraint  to  a  curve 
can  be  imagined  as  due  to  a  narrow  tube  having  the  shape 
of  the  curve,  or  by  imagining  the  particle  as  a  bead  sliding 
along  a  wire. 

In  these  cases  the  constraint  is  complete.  But  it  is  easy 
to  imagine  incomplete,  i.  e.  partial  or  one-sided,  constraints 
of  various  kinds.  Thus  the  rails  compel  a  train  to  follow  a 
definite  curve,  but  they  do  not  prevent  it  from  being  hfted 
off  the  track;  a  stone  attached  to  a  cord  and  swung  around 
by  the  hand  is  not  completely  constrained  to  the  surface  of  a 
sphere,  but  only  prevented  from  passing  outside  of  the  sphere. 

While  complete  constraints  are  generally  expressed  by 
equations,  one-sided  constraints  can  be  expressed  by  in- 
equalities. Thus,  for  the  stone,  the  condition  is  that  its 
distance  r  from  the  hand  cannot  become  greater  than  the 
length  I  of  the  cord :  r  ^l.  As  soon,  however,  as  r  becomes 
less  than  I,  the  constraining  action  ceases  and  the  stone 
becomes  free.  For  this  reason  it  is  in  general  sufficient  to 
consider  constraining  equations;  but  the  nature  of  the  con- 
straint, whether  complete  or  partial,  must  be  taken  into 
account  to  determine  when  and  where  the  constraint  ceases 
to  exist. 

330.  It  is  often  convenient  to  replace  the  constraining 
conditions  by  introducing  certain  forces,  called  reactions  of 
the  constraining  surface  or  curve  (comp.  Art.  232).  Thus, 
in  the  case  of  the  stone  attached  to  the  cord,  we  may  imagine 
the  cord  cut  and  its  tension  introduced,  to  make  the  stone 
free. 

If  the  constraints  are  thus  replaced  by  the  correspbnding 
reactions,  these  unknown  forces  must  be  combined  with  the 


250  KINETICS  [33 1'. 

given  forces,  and  then  the  equations  of  motion  of  a  free 
particle  can  be  used.  Thus,  let  X,  Y,  Z  be  the  components 
of  the  resultant  given  force  F,  X',  Y',  Z'  those  of  the  resultant 
reaction  F';  then  the  equations  of  motion  are 

mx  =  X  -\-  X',     my  =  Y  +  Y',    mz  =  Z  i-  Z' .      (3) 

In  many  applied  problems  the  determination  of  these 
unknown  reactions  is  more  important  than  that  of  the  actual 
motion.  The  term  Kiiietostatics  has  recently  been  proposed 
for  this  branch  of  mechanics. 

2.  Motion  on  a  fixed  curve. 

331.  Let  us  resolve  the  given  force  F  and  the  constraining 
force  F'  each  into  a  tangential  component  Ft,  F/  and  a  com- 
ponent Fn,  Fn  in  the  normal  plane.  The  normal  component 
Fn  of  the  constraint  is  generally  denoted  by  A'^  and  called 
the  normal  reaction  of  the  curve ;  a  force  —  N,  equal  and  op- 
posite to  it,  represents  the  normal  pressure  exerted  by  the 
particle  on  the  curve.  The  tangential  component  F/  of  the 
constraint  exists  only  if  the  curve  is  "  rough,"  i.  e.  offers 
frictional  resistance;  denoting  the  coefficient  of  friction  by 
M  we  have  (Art.  238)  Ft'  =  fiN. 

Hence  the  equations  of  motion  are: 

mi)  =  Ft  -  nN,     m—=  res •  (F„,  N).  (4) 

P 

The  former  of  these  equations  determines  the  actual  motion 
along  the  given  curve.  The  latter  states  that  the  forces 
Fn  and  N  in  the  normal  plane  must  have  a  resultant  along 
the  principal  normal,  toward  the  center  of  curvature,  of 
magnitude  mv^/p ;  this  resultant  is  called  the  centripetal  force. 
A  force   —  mv^lp,  equal  and  opposite  to  this  resultant,  is 


333.]  CONSTRAINED  MOTION  OF  A  PARTICLE  251 

called  centrifugal  force;  it  should  be  noticed  that  this  is  a 
force  exerted  not  on  the  moving  particle,  but  hy  it. 

332.  By  the  second  of  the  equations  (4),  the  centripetal 
force,  mv^/p,  is  the  resultant  of  the  given  normal  force  F„ 
and  the  normal  reaction  N  of  the  curve;  see  Fig.  78  whose 


plane  is  the  normal  plane  of  the  curve,  P  being  the  position 
of  the  particle  and  C  the  center  of  curvature. 

It  follows  that  the  pressure  on  the  curve,  —  N,  is  the  resultant 
of  the  given  normal  force  Fn  and  the  centrifugal  force  —  mv^/p. 

If  in  particular  the  given  force  Fn  is  zero,  or  at  least  negli- 
gible, as  is  often  the  case,  the  pressure  on  the  curve  is  equal 
to  the  centrifugal  force. 

333.  Denoting  by  Nx,  Ny,  N^  the  components  of  the 
normal  reaction  N  and  observing  that  the  frictional  resist- 
ance fxN  is  directed  along  the  curve  opposite  to  the  sense  of 
the  motion  we  find  that  the  equations  (3)  here  assume  the 
form 

mx  =  X  -\-  Nx  —  M-^;p> 

my=  Y  +  Ny-^Nf^,  (5) 

dz 

mz  =  Z  -\-  Nz  —  p.N  -T-, 
ds 

where    N""  =  N,^  -\-  Ny""  +  A^.^    ^nd    NM  +  N ydy  +  N,dz 


252  KINETICS  [334. 

=  0  since  N  is  normal  to  the  path.  In  addition,  we  have  of 
course  the  equations  (2)  of  the  curve. 

Multiplying  the  equations  (5)  by  dx,  dy,  dz  and  adding 
we  find  the  equation  of  kinetic  energy  and  work 

dilmv^)  =  Xdx  +  Ydij  +  Zdz  -  fiNds. 

This  relation  might  have  been  written  down  directly  by  con- 
sidering that  for  a  displacement  ds  along  the  fixed  curve  the 
normal  reaction  N  does  no  work,  while  the  work  of  friction 
is  —  fxNds. 

If  there  be  no  friction  (fi  =  0)  it  follows  from  the  last 
equation,  or  from  the  first  of  the  equations  (4),  that  the 
velocity  is  independent  of  the  reaction  of  the  curve. 

334.  Exercises. 

(1)  A  mass  of  2  lbs.  attached  to  a  cord,  3  ft.  long,  is  swung  in  a 
circle.  Neglecting  gravity,  find  the  tension  in  pounds:  (a)  when  the 
mass  makes  one  revolution  per  second;  {b)  when  it  makes  S  revolutions 
per  second,  (c)  If  the  cord  cannot  stand  a  tension  of  more  than  300 
lbs.,  what  is  the  greatest  allowable  number  of  revolutions? 

(2)  A  plummet  is  suspended  from  the  roof  of  a  railroad  car;  how 
much  will  it  be  deflected  from  the  vertical  when  the  train  is  running 
45  miles  an  hour  in  a  curve  of  300  yards  radius? 

(3)  A  body  on  the  surface  of  the  earth  partakes  of  the  earth's  daily 
rotation  on  its  axis.  The  constraint  holding  it  in  its  circular  path  is  due 
to  the  attractive  force  of  the  earth.  Taking  the  earth's  equatorial  radius 
as  3963  miles,  show  that  the  centripetal  acceleration  of  a  particle  at  the 
equator  is  about  ^  ft.  per  second,  or  about  j^-^  of  the  actually  observed 
acceleration  g  =  32.09  of  a  body  falling  in  vacuo. 

(4)  If  the  earth  were  at  rest,  what  would  be  the  acceleration  of  a 
body  falling  in  vacuo  at  the  equator? 

(5)  Show  that  if  the  velocity  of  the  earth's  rotation  were  over 
17  times  as  large  as  it  actually  is,  the  force  of  gravity  would  not  be 
sufficient  to  detain  a  body  near  the  surface  at  the  equator  (comp.  Ex. 
(13),  Art.  320). 

(6)  Show  that  in  latitude  <f>  the  acceleration  of  a  falling  body,  if 


335. 


CONSTRAINED  MOTION  OF  A  PARTICLE 


253 


B 


the  earth  were  at  rest,  would  hegi  =  g  +  j  cos-(p,  where  g  is  the  observed 
acceleration  of  a  falling  body  on  the  rotating  earth  and  j  the  centripetal 
acceleration  at  the  equator.  Thus,  in  latitude  0  =  45°,  g  =  980.6  cm. ; 
hence  gi  =  982.3. 

(7)  Owing  to  the  earth's  rotation  on  its  axis  the  direction  of  a 
plumb-line  does  not  pass  through  the  center  of  the  earth,  even  when 
the  earth,  as  here  assumed,  is  regarded  as  a  homogeneous  sphere. 
Determine  the  angle  S  of  the  deviation  in  latitude  <P;  in  what  latitude 
is  8  greatest? 

(8)  A  chandelier  weighing  80  lbs.  is  suspended  from  the  coiling 
of  a  hall  by  means  of  a  chain  12  ft.  long  whose  weight  is  neglected. 
By  how  much  is  the  tension  of 
the  chain  increased  if  it  be  set 
swinging  so  that  the  velocity  at 
the  lowest  point  is  6  ft.  per  sec- 
ond? 

335.  A  particle  of  mass  ?n  sub- 
ject to  gravity  alone  is  constrained 
to  move  in  a  vertical  circle  of  ra- 
dius I.  If  there  be  no  friction 
on  the  curve  and  the  constraint 
be  produced  by  a  weightless  rod 
or  cord  joining  the  particle  to 
the  center  of  the  circle,  we  have 
the  problem  of  the  simple  mathe- 
matical pendulum. 


n/^ 

R 

^\N 

f 

o- 

K 

\ 

N 

V                             7 

r\ 

-T^'P 

U 

/ 

V^ 

\j/ 

Fig.  79. 


The  first  of  the  equations  (4),  Art.  331,  is  readily  seen  to  reduce  in  this 
case  (see  Fig.  79)  to  the  form 

Z-^^,+r/sm^=0. 
A  first  integration  gives,  as  shown  in  Kinematics  (Arts.  63,  64), 


hv-'  =  g{l  cose +^    -  IcosO,), 

where  Vo  is  the  velocity  which  the  particle  has  at  the  time  t  =  0  when 
its  radius  makes  the  angle  AOPo  =  do  with  the  vertical.  Multiplying 
by  TO,  we  have,  for  the  kinetic  energy  of  the  particle, 


254  KINETICS  (336. 

imv^  =  777g{l  cos9  +  h), 
where  h  =  Vo^/2g  —  I  cos^o  is  a  constant.     If  the  horizontal  line  MN, 
drawn  at  the  height  vo'^j2g  above  the  initial  point  Po,  intersect  the 
vertical  diameter  AB  at  R,  it  appears  from  the  figure  that  h  =  RO. 

336.  Taking  R  as  origin  and  the  axis  of  z  vertically  downwards^ 

we  have  RQ  =  2  =  I  cos9  +  h;   hence  the  force-function   U  has  the 

simple  expression 

U  =  nigz] 

and  the  velocity  v  =  V2gz  is  seen  to  become  zero  when  the  particle 
reaches  the  horizontal  line  MN. 

For  the  further  treatment  of  the  problem,  three  cases  must  be 
distinguished  according  as  this  line  of  zero-velocity  MA''  intersects 
the  circle,  touches  it,  or  does  not  meet  it  at  all;  i.  e.  according  as 

h  =  l,OT  ~  =  21  cosH^o. 

337.  The  second  of  the  equations  (4),  Art.  331,  serves  to  determine 
the  reaction  N  of  the  circle,  or  the  pressiu-e  —  N  on  the  circle.     We  have 

m  J   =  —  »ig  cosO  +  A'', 
whence 

A^  —  7nl    ,  +  9  cos9    j  . 

Substituting  for  v^  its  value  from  Art.  335,  we  find 

N  =  vig(  2  J  +  3  cosd  )  . 

The  pressiu-e  on  the  curve  has  therefore  its  greatest  value  when  0  =  0, 
i.  e.  at  the  lowest  point  A.  It  becomes  zero  for  I  cosOi  =  —  ^h, 
which  is  easily  constructed. 

338.  If  the  constraint  be  complete  as  for  a  bead  sliding  along  a 
circular  wire,  or  a  small  ball  moving  within  a  tube,  the  pressure  merely 
changes  sign  at  the  point  9  =  9i.  But  if  the  constraint  be  one-sided, 
the  particle  may  at  this  point  leave  the  circle.  The  one-sided  constraint 
may  be  such  that  OP  ^  /,  as  when  the  particle  runs  in  a  groove  cut 
on  the  inside  of  a  ring,  or  when  it  is  joined  to  the  center  by  a  cord;  in 
this  case  the  particle  may  leave  the  circle  at  some  point  of  its  upper  half. 
Again,  the  one-sided  constraint  may  be  such  that  OP  ^  I,  as  when 


340.1  CONSTRAINED  MOTION  OF  A  PARTICLE  255 

the  particle  runs  in  a  groove  cut  on  the  rim  of  a  disk;  in  this  case  the 
particle  can  of  course  only  move  on  the  upper  half  of  the  circle. 

339.  Exercises. 

(1)  For  Oo  =  60°,  I  =  1  ft.,  ^0  =  9  ft.  per  second,  show  that  the  par- 
ticle will  leave  the  circle  very  nearly  at  the  point  di  =  120°,  if  the  con- 
straint be  such  that  OP  ^  I  (Art.  338). 

(2)  For  vo  =  10  ft.  per  second,  everything  else  being  as  in  Ex.  (1) 
show  that  the  particle  will  leave  the  circle  at  the  point  di  =  134^°, 
nearly. 

(3)  A  particle,  subject  to  gravity  and  constrained  to  the  inside  of 
a  vertical  circle  {OP  ^  I),  makes  complete  revolutions.  Show  that  it 
cannot  leave  the  circle  at  any  point,  if  th  >  I;  and  that  it  will  leave 
the  circle  at  the  point  for  wliich  cosO  =  —  ^h/l,  if  §/i  <  I. 

(4)  A  particle  subject  to  gravity  moves  on  the  outside  of  a  vertical 
circle;  determine  where  it  will  leave  the  circle:  (a)  if  MN  (Fig.  79) 
intersects  the  circle;  (h)  if  MN  touches  the  circle;  (c)  if  MN  does  not 
meet  the  circle. 

(5)  A  particle  subject  to  gravity  is  compelled  to  move  on  any 
vertical  curve  z  =  fix)  without  friction.  Show  that  the  velocity  at 
any  point  is  ?;  =  V2gz  (comp.  Art.  336)  if  the  horizontal  axis  of  x  be 
taken  at  a  height  above  the  initial  point  equal  to  the  "height  due  to 
the  initial  velocity,"  i.  e.  Vo^/2g. 

(6)  A  particle  slides  on  the  outside  of  a  smooth  vertical  circle, 
starting  from  rest  at  the  highest  point  of  the  circle.  Find  where  it 
will  meet  the  horizontal  plane  through  the  lowest  point  of  the  circle. 

340.  If  for  a  particle  constrained  to  a  curve,  under  given 
forces,  the  time  of  reaching  any  particular  point  0  is  the 
same  from  whatever  point  of  the  curve  the  particle  starts 
with  zero  velocity,  the  curve  is  called  a  tautochrone  for  the 
given  forces,  and  the  point  0  is  called  the  point  of  tauto- 
chronism. 

In  a  vertical  plane,  if  gravity  is  the  only  force,  a  cycloid 
with  vertical  axis  can  be  shown  to  be  a  tautochrone,  with 
the  vertex  as  point  of  tautochronism.  This  will  even  be 
true  if  the  curve  be  rough,  or  if  the  particle  be  subj-ect  to  a 


250  KINETICS 


[341. 


resistance  proportional  to  the  velocity  in  the  direction  of 
motion;  but,  for  the  sake  of  simplicity,  we  exclude  these 
complications.     . 

The  problem  of  determining  a  tautochrone  for  given  forces 
(if  such  a  curve  exists)  is  rather  different  in  nature  from  the 
ordinary  problems  of  mechanics  inasmuch  as  it  is  here 
required  to  find  a  curve,  on  which  motions  of  a  certain  kind 
may  take  place.  Indeed,  it  is  a  generalization  of  the  problem 
of  the  tautochrone  that  led  Abel  to  the  first  solution  of  an 
integral  equation.* 

341.  With  respect  to  a  horizontal  axis  Ox  and  a  vertical 
axis  Oz  through  the  point  of  tautochronism,  the  principle  of 
kinetic  energy  and  work  (comp.  Art.  339,  Ex.  5)  gives  for  the 

velocity 

v'^  =  2g{h  -  z), 

where  h  is  the  ordinate  of  the  starting  point  P.  Counting 
the  arc  s  from  0  we  have  dsjcit  =  —  V2g(/i  —  z),  whence 
the  time  of  motion  from  P  to  0 : 


^  "  ~  J.=/.  V2sf(/i  -  z)  ^  ^2g  X 


ds 


VT- 


If  we  put  s  =  f{z)  and  hence  ds  =  f{z)dz,  the  problem  re- 
quires the  determination  of  the  function  f{z)  for  which  the 
integral  has  a  value  independent  oi  h.  To  make  the  limits 
independent  of  h  let  us  put  z  =  hy;  we  then  find 


t  = 


1     r'rihy)hdy_    1     r'  ri^y 

This  integral   will   be  independent  of   h  if  j'Qiy)  -yjhy  is 

*See  M.  BocHER,  Integral  equations,  Cambridge,  University  Press, 
1909,  p.  6. 


342.]  CONSTRAINED   MOTION  OF  A  PARTICLE  257 

independent  of  h;  and  as  this  expression  is  symmetric  in  h 
and  y,  it  will  then  be  also  independent  of  z.  We  can  therefore 
put 

whence 


solving  for  dx  we  find  (comp.  Art.  20) : 


X  =    I     -^ dz  =  -^z^K  —  z)  +  K-  sin~i  .*l-  . 

This  is  the  equation  of  a  cycloid  with  0  as  vertex  and  Oz  as 
axis.  Putting  z  =  k  sin^i^,  we  find  the  equations  of  the 
cycloid  in  the  form 

X  =  ^K{e  +  sin^),     z  =  ^k(1  -  cos9), 

so  that  K  is  the  diameter  of  the  generating  circle. 
For  the  time  we  find : 

^  _      i-^     r       dy 


"ViX 


4y  -  y''         ^'2g* 
342.  Exercises. 

(1)  For  a  heavy  particle  moving  witliout  friction  on  a  cycloid  with 
vertical  axis,  x  =  a(d  -\-  sin9),  z  =  a(l  —  cos^),  show  tliat  the  equation 
of  motion  is  s  =  —  gs/4:a,  s  being  the  arc  counted  from  the  vertex. 
Hence,  if  ?>  =  0  for  s  =  so,  s  =  Sa  cos  Vgl^a  t,  which  shows  that  the 
time  of  reaching  the  lowest  point  is  independent  of  so. 

(2)  The  involute  of  a  cycloid  being  an  equal  cycloid,  with  its  vertex 
at  the  cusp,  its  cusp  on  the  axis,  of  the  original  cycloid,  the  particle 
in  Ex.  (1)  can  be  constrained  to  the  cycloid  by  means  of  a  cord  of 
length  2a,  attached  to  the  cusp  of  the  involute,  and  wrapping  itself 
on  a  cylinder  erected  on  the  involute  as  base  {cydoidal  pouhdutii). 
Show  that,  if  the  particle  starts  from  rest  at  the  cusp  of  the  original 
cycloid,  the  tension  of  the  cord  is  twice  the  normal  component  of  the 
weight  of  the  particle. 

18 


258  KINETICS  [343. 

(3)  Prove  that  it  is  not  possible  to  construct  a  tautochrone  (for 
gravity)  from  P  to  0  with  0  as  point  of  tautochronism  unless  the  slope 
of  OA  is  in  absolute  value  <  2/7r. 

343.  The  cycloid  (with  vertical  axis)  has  another  remarkable  prop- 
erty; it  is  the  brachistochrone,  or  cui've  of  quickest  descent,  for  a 
particle  subject  to  gravity.  More  definitely:  two  points  Pi,  P2  being 
given  we  may  inquire  to  what  curve  in  their  vertical  plane  must  a 
heavy  particle  be  constrained  to  reach  in  the  shortest  time  the  lower 
point  P2  if  it  starts  from  Pi  with  a  given  velocity. 

As  the  time  is  given  by  a  definite  integral  the  problem  requires  the 
determination  of  that  curve  z  =  fix)  for  which  this  integral  becomes  a 
minimum.  This  problem  has  given  rise  to  the  invention  of  the  calculus 
of  variations. 

As  the  problem  can  hardly  be  solved  satisfactorily  without  using 
the  methods  of  this  calculus  we  merely  state  that  the  required  curve  is 
the  cycloid  through  the  two  points,  without  cusp  between  them  and 
with  vertical  axis.* 

3.  Motion  on  a  fixed  surface. 

344.  The  equations  of  motion  of  a  particle  constrained  to 

a  surface  do  not  differ  in  form  from  tlie  equations  (5),  Art. 

333,   for  a  particle  constrained  to  a  curve.     The  normal 
reaction 


being  normal  to  the  given  surface  (p(x,  y,  z)  =  0,  we  have 

dip        dip       dip 
dx        dy         dz 

A  comparatively  simple  problem  is  that  of  the  conical  or 
spherical  pendulum,  i.  e.  of  a  particle  subject  to  gravity  and 
constrained  to  the  surface  of  a  sphere.  But  even  this 
problem  can  not  be  treated  without  introducing  elliptic 
integrals. 

*See  O.  BoLZA,  Variationsrechnung,  Leipzig,  Teubner,  1909,  p.  207. 


346.]  CONSTRAINED  MOTION  OF  A  PARTICLE  259 

4.  The  method  of  indetermmate  multipliers. 

345.  The  following  brief  discussion  of  the  equations  of 
motion  of  a  constrained  particle  is  not  so  much  intended  to 
furnish  methods  for  solving  particular  problems,  but  rather 
as  a  preparation  for,  and  an  introduction  to,  the  general 
methods  of  mechanics  of  systems  of  particles  subject  to 
conditions. 

For  this  reason  we  shall  here  assume  the  absence  of  friction 
on  the  constraining  surface  or  curve;  but,  on  the  other  hand, 
it  is  desirable  to  generalize  by  assuming  that  the  constraints 
are  variable,  that  is,  that  the  conditional  equations  (1)  and 
(2),  Art.  328,  contain  the  time  t  explicitly. 

346.  D'Alembert's  Principle.  The  ordinary  equations  of 
motion  of  a  free  particle, 

mx  =  A^,     my  —  Y,     mz  —  Z,  (6) 

where  X,  Y,  Z  are  the  components  of  the  resultant  R  of  the 
given  forces,  mejiely  express  the  equality  of  this  force  R, 
as  a  vector,  to  the  mass-acceleration  mj,  which  is  sometimes 
called  the  effective  Jorce.  It  follows  that  //  the  reversed  effective 
force  —  mj,  or  its  components  —  mx,  —  my,  —  m'z,  he  combined 
with  the  given  forces  we  have  a  system  in  equilibrium  at  the 
given  instant.  This  is  the  fundamental  idea  of  d'Alembert's 
principle,  as  it  is  now  generally  used. 

Owing  to  this  idea  we  can  apply  to  kinetic  problems  the 
statical  conditions  of  equilibrium.  Thus,  in  the  case  of 
the  free  particle,  the  conditions  of  equilibrium  of  the  forces 
X,  Y,  Z,  —  mx,  —  my,  —  mz  are 

X  —  mx  =  0,      Y  —  mij  =  0,     Z  —  mz  —  0, 

and  thus  the  equations  of  motion  arc  found. 

But  the  conditions  of  equilibrium  can  also  be  expressed 


260  KINETICS  [347. 

by  means  of  the  principle  of  virtual  work.  By  Art.  266,  the 
necessary  and  sufficient  condition  of  equiUbrium  of  the 
particle  under  the  forces  —  77ix.  —  mij,  —  m'z,  X,  Y ,  Z  is  that 

(-  mx  +  X)bx  +  (-  vtij  +  r)5?/  +  (-  mz  +  Z)52  =  0  (7) 

for  any  virtual  displacement  bs{bx,  by,  8z).  Owing  to  the 
independence  of  8x,  dy,  8z,  their  coefficients  must  vanish 
separately,  and  we  find  again  the  equations  (6).  In  other 
words,  the  single  equation  (7)  is  equivalent  to  the  three 
equations  (6). 

347.  One  constraint.     If  the  particle  is  subject  to  the 
condition  or  constraint 

ifix,  y,  z,  0=0,  (8) 

it  must  throughout  its  motion  remain  on  the  surface  repre- 
sented by  this  equation.  To  apply  d'Alembert's  prin- 
ciple let  the  particle  be  subjected  to  a  virtual  displacement 
ds.  If  this  displacement  be  selected  along  the  position  of 
the  surface  at  the  time  t,  the  work  of  the  reaction  (which 
is  normal  to  the  surface  (8),  and  hence  to  8s,  since  we  assume 
that  there  is  no  friction)  will  be  zero.  Hence  the  equation 
of  motion  is  the  same  as  for  a  free  particle,  viz.  (7).  But 
the  displacement  5s  must  be  along  the  surface  (8),  or  as 
we  shall  say,  compatible  vrith  the  constraint.  This  requires 
that  8x,  8y,  8z  be  selected  so  as  to  satisfy  the  relation 

<Px8x  +  <p^8y  +  ^^8z  =  0,  (9) 

where  the  partial  derivatives  ^i,  (py,  (p^  of  ip  with  respect  to 
x,  y,  z  are  calculated  regarding  t  as  constant  since  we  want 
a  displacement  along  the  position  of  the  surface  (8)  at  the 
time  t. 

The  equations   (7)   and   (9)   constitute  the  equations  of 
motion  of  the  particle  on  the  surface   (8).     By  means  of 


349.]  CONSTRAINED  MOTION  OF  A  PARTICLE  261 

(9)  one  of  the  component  displacements  8x,  by,  bz  can  be 
eliminated  between  the  two  equations;  the  remaining  two 
displacements  being  arbitrary,  the  two  equations  of  motion 
are  found  by  equating  to  zero  the  coefficients  of  these  two 
chsplacements. 

348.  To  perform  this  elimination  systematically  the 
method  of  indeterminate  multi-pliers  may  be  used  as  follows. 
Multiplying  the  conditional  equation  (9)  by  an  indeter- 
minate multiplier  X  and  adding  the  resulting  equation  to 
(7)  we  find: 

(—  mx  -\-  X  -\r  \(px)8x  +  (—  7ny  -\-  Y  +  '^<Py)8y 

+  (-  VIZ  +  Z  +  \<p,)8z  =  0. 

The  arbitrary  multiplier  X  can  be  selected  so  as  to  make 
the  coefficient  of  any  one  of  the  three  displacements  vanish; 
the  other  two  displacements  being  arliitrary,  their  coeffici- 
ents must  also  vanish.  Hence  the  last  equation  is  equiva- 
lent to  the  three  equations, 

inx  =  X  -\-  X<px,     my  =  Y  +  X^j,,     mz  =  Z  -\-  X^^,  (10) 

which,  in  connection  with  the  given  condition  (8),  are  suf- 
ficient to  determine  x,  y,  z,  and  X  as  functions  of  t. 

349.  By  comparing  (10)  with  (3),  Art.  330,  it  appears 
that 

A      =    \ipxy         1       =    ^fy}       Z      =    \(Pzj 

so  that  the  normal  reaction  is 


N  =  Xi/<^/-+  <p/-+  <p^  (11) 

If  we  combine  the  equations   (10)   by  the  principle  of 
kinetic  energy  and  work,  we  find 

dihnv"^)  =  Xdx  +  Ydy  +  Zdz  -\-  \{(pxdx  +  ^ydy  +  <pdz). 


262  KINETICS  [350. 

Here  the  elementary  work  which  constitutes  the  right- 
hand  member  contains  in  general  terms  depending  on  the 
reaction.  This  is  due  to  the  fact  that  the  displacement 
ds{dx,  dy,  dz)  here  used  is  along  the  moving  or  variable 
surface  (8),  and  not  along  its  position  at  the  time  t. 

If  the  surface  (8)  be  fixed  we  have  of  course  (p^dx  +  (pjjdy 
+  ^zdz  =  0  so  that  the  equation  reduces  to 

d{hnv'')  =  Xdx  +  Ydy  +  Zdz. 

In  the  general  case,  since  ifxdx  +  tpydy  -\-  ipzdz  -\-  cptdt  =  0, 
the  equation  of  kinetic  energy  and  work  can  be  written 

di^mv^)  =  Xdx  +  Ydy  +  Zdz  -  X^ptdt.  (12) 

350.  Two  constraints.     If  the  particle  be  subject  to  two 

conditions 

ip{x,  y,  z,  t)  =  0,     4^(x,  y,  z,  t)  =  0  (13) 

it  will  move  along  the  curve  of  intersection  of  the  surfaces 
represented  by  these  equations. 

For  a  displacement  8s  along  the  position  of  this  curve 
at  the  time  t  the  work  of  the  reaction  is  again  zero  so  that 
the  general  equation  (7)  holds  for  such  a  displacement. 
To  obtain  such  a  displacement  we  must  subject  its  compo- 
nents 8x,  8y,  8z  to  the  conditions 

(pjx  +  (py8y  +  (p,8z  =  0,     \px8x  +  \py8y  +  \p,8z  =  0.  (14) 

Between  the  three  equations  (7)  and  (14)  two  of  the  dis- 
placements 8x,  8y,  8z  can  be  eliminated,  and  the  coefficient 
of  the  third  equated  to  zero  gives  the  equation  of  motion 
along  the  curve  (13). 

351.  To  perform  this  elimination  in  a  systematic  way, 
multiply  (14)  by  indeterminate  multipliers  X,  n  and  add 
to  (7).     In  the  resulting  equation 


352.1  CONSTRAINED   MOTION  OF  A  PARTICLE  263 

(—  mx  +  X  +  Xv7x  +  )U'/'x)5x  +  (-  my  +  F  +  X.^^  +  ix^py)by 

+  ( -  mz  +  Z  +  X<^,  +  ^x^p,)bz  =  0 

the  arbitrary  multipliers  X,  /x  can  be  selected  so  that  the 
coefScients  of  two  of  the  displacements  bx,  by,  bz  vanish; 
and  then  the  coefficient  of  the  third  must  also  vanish.  Thus 
we  find  the  three  equations  of  motion, 

mx  =X  -\-  \(p^  +  /xi/'x,     my  =  Y  +  \^y  +  /x^/'^,, 

mz  =  Z  +  X<^,  +  ix^„  ^^^^ 

which,  together  with  the  conditions  (13),  are  sufficient  to 
determine  x,  y,  z,  \,  jjl  as  functions  of  t. 

5.  Lagrange's  equations  of  motion. 

352.  Generalized  Co-ordinates.  To  determine  the  posi- 
tion of  a  point  P  in  space  we  may  use,  instead  of  the  cartesian 
co-ordinates  x,  y,  z,  a  large  variety  of  other  systems  of  co- 
ordinates, e.  g.  polar  or  spherical,  cylindrical  (Art.  56,  Ex.  9), 
elliptic  (Arts.  408,  411)  co-ordinates,  etc.  Indeed,  any  three 
linearly  independent  functions  of  x,  y,  z,  say 

gi  =  qi{x,  y,  z),     go  =  qo(x,  y,  z),     qz  =  Qzix,  y,  z), 

can  be  taken  as  such  generalized,  or  lagrangian,  co-ordinates 
of  P,  at  least  within  a  certain  region  of  space.  Each  of  these 
functions  equated  to  a  constant  represents  a  surface,  and 
the  point  P(x,  y,  z)  is  determined  as  intersection  of  the  three 
surfaces. 

Solving  these  equations  for  x,  y,  z  we  find  x,  y,  z  as  functions 
of  gi)  g2,  g3-  For  the  sake  of  generality  we  shall  assume  that 
X,  y,  z.  are  given  as  functions  of  gi,  go,  ga,  and  of  the  time  t: 

X  =  x(q^,go,q3,t),    y  =  y{q\,  q^,  qz,  t) ,    z  =  z(q^,q2,qz,t),    (16) 


264  KINETICS  [353. 

SO  that  the  new  system  of  co-ordinates  is  a  moving  or  variable 
system. 

By  using  such  generalized  co-ordinates  and  introducing  the 
kinetic  energy  T  and  its  derivatives  the  equations  of  motion 
of  a  particle  with  or  without  constraints  can  be  put  into  a 
remarkably  compact  form  which  was  first  devised  by  La- 
grange for  the  general  equations  of  motion  of  a  system  of  n 
particles  (comp.  Chap.  XX). 

353.  Free  Particle.  By  multiplying  the  ordinary  equa- 
tions of  motion 

mx  =  X,    my  =  Y,    mz  =  Z 
hy  dx/dqi,  dy/dqi,  dz/dqi  and  adding  we  find 

/  ..  dx     ,     ..  dy     ,    ..  dz\        ^dx     ,    ^^  dy     ,    „  dz 

mix—  +  y^~  +  2;r~     =X- VY  ^  +^J"- 

\     dqi        ''dqi         dqi  /  dqi  dqi  dqi 

The  right-hand  member  we  shall  denote  briefly  by  Qii 

dqi  dqi  dqi 

this  Qi  may  be  called  the  generalized  force  corresponding  to 
the  co-ordinate  gi  (comp.  Art.  354). 

The  main  point  lies  in  the  transformation  of  the  left-hand 
member.  Consider  the  first  term  in  the  parenthesis;  by  the 
formula  for  the  differentiation  of  a  product  we  have  the 
identity  q^        d  /  .  dx\        .  dx 


*  dqi        dt  \     dqi)  dqi 

Treating  the  other  two  terms  in  the  same  way  we  find  that 
our  equation  can  be  written: 

d  (  .  dx    ,    .  dy    ,    .  dz 
dt  \     dqi        -^  dqi  dqi 

/  .  dx    ,     .  dy     ,    .  dz  ,        „  .._v 

-mix^  +y-^  +z--)  =Qi,      (17) 
V     dqi        -"dqi  dqi' 


354.]  CONSTRAINED  MOTION  OF  A  PARTICLE  265 

where  the  second  term  is  evidently  the  gi-derivative  of  the 
kinetic  energy 

T    =    iw(x2   ^   ^^2   ^   ^2)_ 

To  interpret  the  first  term  observe  that  the  equations  (16) 
give 

dx   .     ,   dx    .     ,    dx  . 

hence,  if  we  regard  i  as  a  function  of  gi,  52,  93,  qi,  Qi,  qz,  t,  we 
have 

dx    _  dx         dx   _  dx        dx   _  dx 
dqi       dqi^     dq-i       5^2'     dqz       dqs' 

Similar  relations  hold  of  course  for  y  and  z.  We  can  therefore 
in  the  first  term  of  (17)  replace  5a:/6gi,  dy/dqi,  dz/dqi  hydx/dqi, 
dij jdqi,  dz I dqi;  and  then  it  appears  that  this  term  is  equal  to 
the  time-derivative  of  the  gi-derivative  of   T.     Thus  (17) 

becomes 

d  dj^  _dT  ^ 
dt  dqi       dqi 

By  multiplying  the  ordinary  equations  of  motion  by  the 
derivatives  of  x,  y,  z  with  respect  to  52  and  ^3  we  obtain  two 
similar  equations.     Thus  Lagrange's  equations  of  motion  for 
.  a  free  particle  are : 

dtdqi      dqi       ^''     ~dtdq2      dq^       ^''      dt  dqz       dqs      ^"  ^     ^ 

354.  If  there  exists  a  force-function  U  for  the  forces  X,  Y, 

Z,  I.  e.  if 

^_dU       y  _djl  dU 

^  ~  dx'  dy'     ^        dz' 

we  have 

^dJJd^      dJJ^y       dJUdz^^dU 

^'  ~  dx  dqi  ^  dy  dqi  "^  dz  dqi      dqi ' 


266  KINETICS  (355. 

and  similarly 

_dU  _dU 

In  this  case  one  of  the  three  equations  (18)  can  be  replaced 
by  the  equation  of  kinetic  energy  and  work 

T  =  U  -\-h, 
where  h  is  a  constant. 

355.  Constrained  particle.     In  the  case  of  one  constraint, 

<p{x,  ij,  z,  t)  =  0, 

the  position  of  the  particle  on  this  surface  is  determined  by 
two  co-ordinates  qi,  52;  and  by  applying  the  process  of  Art. 
353  to  the  equations  (10),  Art.  348,  we  find  the  two  equations 
of  motion 

ddT      dT      ^         ddT      dT       „  ,,„,. 

dtdqi      dqi  dtdqo      dq-^ 

For,  the  coefficients  of  X  in  the  right-hand  members,  viz. 

dx  dy  dz  dx  dy  dz 

dqi  dqi  dqi  dq-z  dq-i  dq^ 

are  zero  since  the  particle  moves  on  the  surface  <p  =  0. 
Similarly,  in  the  case  of  two  constraints, 

ip{x,  y,  z,  t)  =  0,     yp{x,  y,  z,  t)  =  0, 

the  position  of  the  particle  on  the  curve  represented  by  these 
equations  is  determined  by  a  single  co-ordinate  q,  and  the 
equation  of  motion  is 

ddT      dT  _ 

dtJi'dq;-^-  ^^^^ 

It  is  obtained  from  the  equations  (15),  Art.  351,  by  the 
process  of  Art.  353.     The  coefficient  of  X,  viz. 


355.]  CONSTRAINED  MOTION  OF  A  PARTICLE  267 

dx  dy  dz 

vanishes  since  it  is  proportional  to  the  cosine  of  the  angle 
made  at  the  instant  considered  by  the  tangent  to  the  con- 
straining curve  with  the  normal  to  the  surface  ^  =  0; 
similarly  for  the  coefficient  of  n. 

The  equations  (18'),  (18")  are  sometimes  distinguished 
from  the  equations  (10),  (15)  as  Lagrange's  equations  of  the 
second  kind,  the  forms  (10),  (15)  being  also  due  to  Lagrange. 


CHAPTER  XV. 
THE  EQUATIONS  OF  MOTION  OF  A  FREE  RIGID  BODY. 

356.  In  kinetics  it  is  convenient  to  think  of  a  rigid  body 
primarily  as  a  finite  number  of  particles  (Art,  156)  connected 
by  a  rigid  framework  without  mass.  The  rigidity  then  con- 
sists on  the  one  hand,  in  the  invariability  of  the  distances 
of  the  particles,  on  the  other  in  the  assumption  (Art.  197) 
that  a  force  applied  to  the  rigid  body,  i.  e.  to  any  one  of  the 
particles,  can  be  imagined  applied  at  any  point  of  its  line 
of  action. 

357.  Consider  any  one  particle  m  of  the  body  and  let  it  be 

cut  loose  from  the  other  particles;  that  is,  let  the  members  of 

the  framework  that  attach  it  to  the  body  be  replaced  by 

tensions  or  pressures.     These  internal  forces,  together  w^th 

the  external  forces  that  may  happen  to  be  applied  at  our 

particle,  will  have  a  resultant  F.     The  equation  of  motion  of 

this  particle  is  therefore 

mj  =  F, 

or,  resolving  along  fixed  rectangular  axes, 

mx  =  X,     my  =  Y,     mz  =  Z.  (1) 

Notice  particularly  that  the  components  X,  Y,  Z  oi  F 
contain  not  only  the  given  external,  but  also  the  unknown 
internal,  forces. 

358.  Such  a  set  of  three  equations  can  be  written  down  for 
each  particle;  hence,  if  the  body  consists  of  n  particles,  there 
would  be  in  all  3n  equations. 

The  number  of  conditions  expressing  the  invariability  of 

the  distances  between  n  particles  is  2>n  —  6.     For  if  there 

268 


359.]       EQUATIONS  OF  MOTION  OF  A  RIGID  BODY         269 

were  but  3  particles,  the  number  of  independent  conditions 
would  evidently  be  3;  for  every  additional  particle,  3  ad- 
ditional conditions  are  required.  Hence,  the  total  number 
of  conditions  is  3  +  3(n  —  3)  —  Sn  —  6. 

It  follows  that  if  a  rigid  body  be  subject  to  no  other  con- 
straining conditions,  the  number  of  its  equations  of  motion 
must  be  Sn  —  (3n  —  6)  =6.  Hence,  a  free  rigid  body  has  six 
independent  equations  of  motion  (comp.  Art.  231).  'h  .  '<  '•  '''■ 

359.  The  six  equations  of  motion  of  the  rigid  body  can  be 
obtained  as  follows. 

Imagine  the  equations  (1)  written  down  for  every  particle, 
and  add  the  corresponding  equations.  This  gives  the  first 
3  of  the  6  equations  of  motion: 

^mx  =  2X,     Zmij  =  2  7,     llmz  =  2Z.  (2) 

It  is  important  to  notice  that  the  internal  reactions  be- 
tween the  particles  which  make  the  body  rigid  occur  in  pairs 
of  equal  and  opposite  forces,  and  form,  therefore,  a  system 
which  is  in  equilibrium  by  itself.  This  may  be  regarded  as 
an  assumption  which  should  be  included  in  the  definition  of 
the  rigid  body.  Hence,  while  these  internal  forces  enter  into 
the  equations  (1),  they  do  not  appear  in  the  equations  (2). 
The  right-hand  members  of  these  equations  (2)  represent 
therefore  the  components  Rx,  Ry,  Rz  of  the  resultant  R  of 
all  the  external  forces  acting  on  the  body.  The  left-hand 
members  can  be  written  in  the  form  d{I,mx)/dt,  d{'Zmij)ldt, 
d{'Zmz)/dt:  these  are  the  time-derivatives  of  the  sums  of  the 
linear  momenta  of  all  the  particles  parallel  to  the  axes.  The 
equations  (2)  can  therefore  be  written  in  the  form 

^Xmx^Rx,     ~i:my  =  Ry,    ^^^^^mz  =  R,.        (2') 

The  axes  of  co-ordinates  are  arbitrary.     Hence,  if  we  agree  to 


270  KINETICS  [360. 

call  linear  momentum  of  the  body  in  any  direction  the  algebraic 
sum  of  the  linear  momenta  of  all  the  particles  in  that  direc- 
tion, the  equations  (2')  express  the  proposition  that  the  rate 
at  which  the  linear  mofnentum  of  a  rigid  body  in  any  direction 
changes  with  the  time  is  equal  to  the  sum  of  the  comyonents  of 
all  the  external  forces  in  that  direction. 

360.  Let  us  now  combine  the  second  and  third  of  the 
equations  (1)  by  multiplying  the  former  by  z,  the  latter  by  y, 
and  subtracting  the  former  from  the  latter.  If  this  be  done 
for  each  particle,  and  the  resulting  equations  be  added,  we 
find  I,m{yz  —  zij)  =  '^{ijZ  —  zY).  Similarly,  we  can  pro- 
ceed with  the  third  and  first,  and  with  the  first  and  second 
of  the  equations  (1).     The  result  is: 

^m{yz  -  zij)  =  Z(yZ  -  zY),     ^m(zx  -  xz)  =  Z{zX  -  xZ), 
^ni{xij  -  yx)  =  Z{xY  -  yX).  (3) 

Here  again  the  internal  forces  disappear  in  the  summation, 
so  that  the  right-hand  members  are  the  components  Hx,  Hy, 
Hz,  of  the  vector  H  of  the  resultant  couple,  found  by  reducing 
all  the  external  forces  for  the  origin  of  co-ordinates.  The 
left-hand  members  are  the  components  of  the  resultant  couple 
of  the  effective  forces  for  the  same  origin. 

We  can  also  say  that  the  right-hand  members  are  the  sums 
of  the  moments  of  the  external  forces  about  the  co-ordinate 
axes  (Art.  229),  while  the  left-hand  members  represent  the 
moments  of  the  effective  forces  about  the  same  axes.  The 
latter  quantities  are  exact  derivatives,  as  shown  in  Art.  279. 
The  equations  (3)  can  therefore  be  written  in  the  form 

y.  'Lm{yz  -  zy)  =  Hx,     ^J^mizx  -  xz)  =  Hy, 

d  ^^'^ 

-^^^mixy  -yx)  =  H^. 


362.]  EQUATIONS  OF  MOTION  OF  RIGID  BODY  271 

As  explained  in  Art.  279,  the  quantity  m{yz  —  zy)  is  called 
the  angular  momentum  (or  the  moment  of  momentum)  of  the 
particle  m  about  the  axis  of  x.  We  shall  now  agree  to  call 
the  quantity  I,m{yz  —  zy)  the  angular  momentu7n  of  the  body 
about  the  axis  of  x,  just  as  Hmx  is  the  linear  momentum  of 
the  body  along  this  axis;  and  similarly  for  the  other  axes. 
The  meaning  of  the  equations  (3')  can  then  be  stated  as 
follows:  The  rate  at  ivhich  the  angular  ynomentum  of  a  rigid 
body  about  any  axis  changes  with  the  time  is  equal  to  the  sum 
of  the  moments  of  all  the  external  forces  about  this  line. 

The  equations  (2)  and  (3),  or  (2')  and  (3'),  are  the  six 
equations  of  motion  of  the  rigid  body.  The  three  equations 
(2)  or  (2')  may  be  called  the  equations  of  linear  momentum, 
while  (3)  or  (3')  are  the  equations  of  angular  momentum. 

361.  If,  as  in  Art.  280,  we  imagine  the  angular  momentum 
of  each  particle  represented  by  a  vector  drawn  from  the 
origin  of  co-ordinates,  the  geometric  sum,  or  resultant,  of 
these  vectors  is  a  vector  h  which  represents  the  angular 
momentum  of  the  body  about  the  origin;  and  its  components 
hx,  hy,  hz  along  the  axes  are  tlie  angular  momenta  1,7n{yz  — 
zy),  'Zim{zx  —  xz),  ^m{xy  —  yx)  of  the  body  about  these 
axes.  The  equations  (3')  can  then  be  written  in  the  simple 
form 

afix        -rj       any       jj       anz       jj  /o//\ 

d^  ==  ^-    7it^  ^^-    ^dt  ^  ^"  ^^  ^ 

and  these  equations  are  together  equivalent  to  the  single 

vector  equation 

dh  _  „ 
dt   ~ 

362.  The  equations  of  linear  momentum,  (2)  or  (2'),  admit 
of  a  further  simplification,  owing  to  the  fundamental  property 


272  KINETICS  [363. 

of  the  centroid.     By  Art.  159,  the  co-ordinates  x,  y,  z  of  the 
^centroid  satisfy  the  relations 

Mx  =  llmx,     My  =  2m2/,     Mz  =  ^mz, 

where  M  =  ^m  is  the  whole  mass  of  the  body.     Differentiat- 
ing these  equations,  we  find 

Mx  =  Imx,     My  =  limy,     Mz   =  2mi, 
and 

Mx  =  Smx,     My  =  2my,     Mz  =  ^mz, 

where  x,  y,  z  are  the  components  of  the  velocity  v,  and  x,  y,  z 
those  of  the  acceleration  j,  of  the  centroid. 

The  equations  (2)  or  (2')  can  therefore  be  reduced  to  the 
form 

Mx  =  j,Mx  =  R.,    Mi)  =  J.  My  =  Ry, 

M2  =  -Mz  =  Rz, 
at 

whence 

Mj=  -^.Mv  =  R; 

i.  e.  if  the  whole  mass  of  the  body  be  regarded  as  concentrated 
at  the  centroid,  the  effective  force  of  the  centroid,  or  the 
time-rate  of  change  of  its  momentum,  is  equal  to  the  resultant 
of  all  the  external  forces.  It  follows  that  the  centroid  of  a 
rigid  body  moves  as  if  it  contained  the  whole  mass,  and  all  the 
external  forces  were  applied  at  this  point  parallel  to  their 
original  directions. 

363.  If,  in  particular,  the  resultant  R  vanish  (while  there 
may  be  a  couple  H  acting  on  the  body),  we  have  by  (2") 
j  =  0;  hence  v  =  const.;  i.  e.  if  the  residtant  force  he  zero 
the  centroid  moves  uniforinly  in  a  straight  line. 


364.]        EQUATIONS  OF  MOTION  OF  A  RIGID  BODY         273 

This  proposition,  which  can  also  be  expressed  by  saying 
that  ii  R  =  0,  the  momentum  Mv  of  the  centroid  remains 
constant,  or,  using  the  form  (2')  of  the  equations  of  motion, 
that  the  hnear  momentum  of  the  body  in  any  direction  is 
constant,  is  known  as  the  principle  of  the  conservation  of 
linear  momentum,  or  the  principle  of  the  conservation  of 
the  motion  of  the  centroid. 

364.  Let  us  next  consider  the  equations  of  angular  momen- 
tum, (3)  or  (3').  To  introduce  the  properties  of  the  centroid, 
let  us  put  X  —  X  =  ^,  y  —  y  =  r],  z  —  z  =  ^,  so  that  ^,  77,  f 
are  the  co-ordinates  of  the  point  (x,  y,  z)  with  respect  to 
parallel  axes  through  the  centroid.  The  substitution  of 
X  =^  X  -\-  ^,  y  =  y  -\-  V,  z  =  z  -{-  ^  and  their  derivatives  in 
the  expression  yz  —  zy  gives 

yz  -  zij  =  yz  -  zy  -\-  ijt  -  zi]  +  -nz  -  ty  +  r]t  -  ^v. 

To  form  'Zm(yz  —  zy)  we  must  multiply  by  m  and  sum 
throughout  the  body;  in  this  summation,  y,  z,  y,  z  are  constant 
and  by  the  property  of  the  centroid,  '^mt]  =  0,  2m^  =  0, 
Sm^  =  0,  2mf  =  0.     Hence  we  find 

'Emiyz  -  zy)  =  2m (17^  -fTJ)  +  M(;yz  -  zy). 

The  second  term  in  the  right-hand  member  is  the  angular 
momentum  of  the  centroid  about  the  axis  of  x  (the  whole  mass 
M  of  the  body  being  regarded  as  concentrated  at  this  point), 
while  the  first  term  is  the  angular  momentum  of  the  body 
(in  its  motion  relatively  to  the  centroid)  about  a  parallel  to 
the  axis  of  x,  drawn  through  the  centroid. 

Similar  relations  hold  for  the  angular  momenta  about  the 

axes  of  y  and  z;  and  as  these  axes  are  arbitrary,  we  conclude 

that  the  angular  momentum  of  a  rigid  body  about  any  li7ie  is 

equal  to  its  angular  momentum  about  a  parallel  through  the 

19 


274  KINETICS  [365. 

centroid  plus  the  angular  momentum  of  the  centroid  about  the 
former  line. 

365.  Differentiating  the  above  expression,  we  find 

-^^m{yz  -  zy)  =  -^^^^Mvt  -  ^v)  +  M(yz  -  zy). 

The  first  of  the  equations  (3')  can  therefore  be  written 

^2m(7?f  -  In)  +  Miyl  -  ztj)  =  H.. 

Now,  if  at  any  time  t  the  centroid  were  taken  as  origin,  so 
that  y  =  0,  z   =0,  this  equation  would  reduce  to  the  form 

J-2m(7?f -N)  =  H., 

which  is  entirely  independent  of  the  co-ordinates  of  the  cen- 
troid. On  the  other  hand,  wherever  the  origin  is  taken,  if 
the  centroid  were  a  fixed  point,  the  same  equation  would 
be  obtained. 

Similar  considerations  apply  of  course  to  the  other  two 
equations  (3')-  It  follows  that  the  motion  of  a  rigid  body 
relative  to  the  centroid  is  the  same  as  if  the  centroid  were  fixed. 

366.  If,  in  particular,  the  resultant  couple  H  be  zero  for 
any  particular  origin  0  (which  will  be  the  case  not  only  when 
all  external  forces  are  zero,  but  whenever  the  directions  of 
all  the  forces  pass  through  the  point  0),  the  equations  (3') 
can  be  integrated  and  give 

i:m{yz  -  zy)  =  d,     i:m{z±  -  xz)  =  Ci,  ... 

Zm(xy  -  yx)  =  C^, 

where  d,  d,  C3  are  constants  of  integration.     Hence,  if  the 

external  forces  pass  through  a  fixed  point,  the  angidar  momentum 
of  the  body  about  any  line  through  this  point  is  constant;  if  there 
are  no  external  forces,  the  angular  momentum  is  constant  for 


368.1  EQUATIONS  OF  MOTION  OF  RIGID  BODY  275 

amj  line  whatever.  This  is  the  principle  of  the  conservation 
of  angular  momentum. 

367.  Taking  the  equations  of  angular  momentum  in  the 
form  (3")  we  find  when  //  =  0; 

/ix  =  Ci,     hy  =  Cs,     h  =  C^,  (4') 

and  hence  the  vector  h  (Fig.  70,  Art.  280)  remains  constant 
in  magnitude  and  direction.  The  term  principle  of  the  con- 
servation of  areas  which  is  often  used  instead  of  principle  of 
the  conservation  of  angular  momentum  is  less  appropriate. 
In  the  case  of  the  single  particle,  where  hx  =  m(yz  —  zy), 
etc.,  the  vector  of  angular  momentum  h  is  simply  2m  times 
the  vector  representing  the  sectorial  velocity;  but  in  the  case 
of  the  rigid  body,  to  form  the  vector  of  angular  momentum  h 
we  have  to  multiply  the  sectorial  velocity  of  each  particle 
by  twice  its  mass  and  add  these  "  weighted"  sectorial  velocities 
geometrically. 

In  the  study  of  the  motion  of  the  rigid  body  with  a  fixed  point 
where  the  vector  h  is  of  primary  importance  it  has  l)ecn 
called  the  impulse,  or  impulse-vector.  Our  principle  then 
means  that  whenever  for  any  point  0  the  resultant  couple  H  is 
zero  the  impulse  remains  a  constant  vector: 

h  =  C. 

The  direction  of  h  is  then  called  the  invariable  direction;  the 
plane  through  0,  perpendicular  to  h, 

Cix  +  Coy  +  Csz  =  0, 
is  called  Laplace's  invariable  plane. 

368.  Returning  to  the  general  case  of  the  motion  of  a  rigid 
body  under  any  forces,  we  may  say  that  the  propositions  at 
the  end  of  Arts.  362  and  3G5  cstal^lish  the  principle  of  the 
independence  of  the  motions  of  translation  and  rotation.     Ac- 


276  KINETICS  [369. 

cording  to  these  propositions  the  problem  of  the  motion  of  a 
rigid  body  resolves  itself  into  two  problems;  that  of  the  mo- 
tion of  the  centroid  and  that  of  the  motion  of  the  body  about 
its  centroid.  The  former  reduces  by  Art.  362  to  the  problem 
of  the  motion  of  a  particle,  viz.  the  centroid  of  the  body,  with 
a  mass  M  equal  to  that  of  the  body,  acted  upon  by  all  the 
given  external  forces  transferred  parallel  to  themselves  to  the 
centroid. 

The  latter  problem,  that  of  the  motion  of  the  bod}^  about 
its  centroid,  is,  by  Art.  365,  the  same  as  the  problem  of  the 
motion  of  a  rigid  body  about  a  fixed  point.  This  important 
problem  is  discussed  in  Chap.  XVIII;  its  solution  depends 
on  the  equations  (3),  (3'),  or  (3")- 

369.  If  the  equation  of  motion  (1),  Art.  357,  of  the  par- 
ticle m  be  multiplied  by  the  components  dx,  dy,  dz  of  the 
actual  displacement  ds  of  this  particle,  we  find  upon  adding 
the  equations  for  all  the  particles 

^m{xdx  +  ijdy  +  zdz)  =  ^(Xdx  +  Ydy  -{-  Zdz), 

where  the  right-hand  member  represents  the  elementary 
work  of  the  external  forces  since  that  of  the  internal  forces  is 
zero.  The  left-hand  member,  just  as  in  the  case  of  the 
single  particle  (Art.  271),  is  the  exact  differential  of  the 
kinetic  energy 

T  =  'Ehnv^  =  i:hn{x^  -f  7J~  +  i') 

of  the  body.  Hence,  integrating,  say  from  t  =  0  to  t  =  t, 
we  find  the  relation 

T  -  To  =  ^hnv^  -  Simvo^*  =  C^iXdx  -f  Ydy  -f  Zdz), 

where  the  right-hand  member  represents  the  work  done  by 
the  external  forces  on  the  body  during  the  time  t.     This 


371.]        EQUATIONS  OF  MOTION  OF  A  RIGID  BODY         277 

equation  expresses  the  principle  of  kinetic  energy  and  work, 
for  a  free  rigid  body:  in  any  motion  of  the  body,  the  increase 
of  the  kinetic  energy  is  equal  to  the  work  done  by  the  external 
forces. 

370.  By  introducing  the  co-ordinates  of  the  centroid, 
i.  e.  by  putting  x  ^  x  -{-  ^,  y  =  y  -\-  v,  z  =  z  -\-  ^,  as  in 
Art.  364,  the  expression  for  the  kinetic  energy  assumes  the 
form  (since  Sm|  =  0,  2m^  =  0,  I,nit  =  0) : 

T  =  7:hn(x'  +  y'  +  -z')  +  ^hn{t~  +  ^-  +  f') 

where  v  is  the  velocity  of  the  centroid  and  u  the  relative 
velocity  of  any  particle  m  with  respect  to  the  centroid. 

Thus,  it  appears  that  the  kinetic  energy  of  a  free  rigid  body 
consists  of  two  'parts,  one  of  which  is  the  kinetic  energy  of  the 
centroid  (the  whole  masss  being  regarded  as  concentrated 
at  this  point),  ivhile  the  other  may  be  called  the  relative  kinetic 
energy  with  respect  to  the  centroid. 

371.  By  the  same  substitution  the  right-hand  member  of 
the  first  equation  of  Art.  369,  i.  e.  the  elementary  work 
Ii(Xdx  +  Ydy  +  Zdz),  resolves  itself  into  the  two  parts 

{dx^X  +  dy^Y  +  dz^Z)  +  2(Xd^  +  Ydv  +  Zd^). 

The  first  parenthesis  contains  the  work  that  would  be  done 
by  all  the  external  forces  if  they  were  applied  at  the  centroid ; 
it  is  therefore  equal  to  the  change  in  the  kinetic  energy  of 
the  centroid,  that  is,  to  d{^Mv^).  The  equation  of  kinetic 
energy  reduces,  therefore,  to  the  following 

dXiimu'')  =  2(Xd^  +  Ydr]  +  Zd^); 

in  other  words,  the  principle  of  kinetic  energy  holds  for  the 
relative  motion  with  respect  to  the  centroid. 


278  KINETICS  [372. 

372.  Impulses.  The  equations  determining  the  effect 
of  a  system  of  impulses  on  a  rigid  body  are  readily  obtained 
from  the  general  equations  of  motion  (2)  and  (3).  We  shall 
denote  the  impulse  of  a  force  F  by  F.  It  will  be  remembered 
that  the  impulse  F  oi  a,  force  F  is  its  time  integral  (Art.  172) ; 
i.  e. 

F  =  j^' fdt. 

We  confine  ourselves  to  the  case  when  t'  —  t  is  very  small 
and  F  very  large,  in  which  case  the  action  of  the  impulsive 
force  F  is  measured  by  its  impulse  F. 

If  all  the  forces  acting  on  a  rigid  body  are  of  this  nature, 
and  the  impulses  of  X,  Y,  Z  during  the  short  interval 
t'  —  t  be  denoted  by  X,  Y,  Z,  the  integration  of  the  equa- 
tions (2)  from  t  =  t  to  t  =  t'  gives 

Sm(a;'  -  x)  =  2X,     ^m (ij'  -  t/)  =  2  Y,     ^m  (i' - i)  =  2Z,    (5) 

where  x,  y,  z  denote  the  velocities  of  the  particle  ?n  at  the 
time  t  just  before  the  impulse,  and  x',  y' ,  z'  those  at  the 
time  t'  just  after  the  action  of  the  impulse. 
Similarly  the  equations  (3)  give 

^m[y{i'  -  i)  -  z{y'  -  ?/)]  =  ^{yZ  -  zY), 

2m[2(i'  -  x)  -  x{z'  -  z)\  =  Z{zX  -  xZ),  (6) 

Xmixiy'  -y)-  y(x'  -  x)]  =  Z{xY  -  yX). 

373.  In  detcrmiming  the  effect  on  a  rigid  body  of  a  system 
of  such  impulses,  any  ordinary  forces  acting  on  the  body  at 
the  same  time  are  neglected  because  the  changes  of  velocity 
produced  by  them  during  the  very  short  time  t'  —  t  are 
small  in  comparison  with  the  changes  of  velocity  x'  —  i, 
y'  —  y,  z'  —  z  produced  by  the  impulses.  If  the  impulse 
F  of  an  impulsive  force  F  be  defined  as  the  limit  of  the  integral 


373.]  EQUATIONS  OF  MOTION  OF  RIGID  BODY  279 

r'Fdt  when  t'  —  t  approaches  zero  and  F  approaches  infinity, 

it  is  strictly  true  that  the  effect  of  ordinary  forces  can  be 
neglected  when  impulsive  forces  act  on  the  body. 

If  the  rigid  body  be  originally  at  rest,  it  will  be  convenient 
to  denote  by  x,  y,  z  the  components  of  the  velocity  of  the 
particle  tn  just  after  the  action  of  the  impulses.  We  may 
also  denote  by  R  the  resultant  of  all  the  impulses,  by  H  the 
resultant  impulsive  couple  for  the  reduction  to  the  origin 
of  co-ordinates,  and  mark  the  components  of  R  and  H  by 
subscripts,  as  in  the  case  of  forces.  With  these  notations  the 
effect  of  a  system  of  impulses  on  a  body  at  rest  is  given  by 
the  equations 

'Zmx  =  Rjr,     ^my  =  Ry,     ^mz  =  R^,  (5') 

I,'m(yz  —  zi/)  =  Hx,     ^m{zx  —  xz)  =  Hy,     Zm{xij  —  yx)  =  H,.  (6') 

In  the  equations  (5')  we  have,  of  course,  Xmx  =  Mx,  I,77iy 
—  My,  l^mz  =  Mz,  where  x,  y,  z  are  the  components  of  the 
velocity  of  the  centroid,  and  M  is  the  mass  of  the  body;  i.  e. 
the  momentum  of  the  centroid  is  equal  to  the  resultant  impulse. 
The  meaning  of  the  equations  (6')  can  be  stated  by  saying 
that  the  angular  momentum  of  the  body  about  any  axis  is  equal 
to  the  moment  of  all  the  impulses  about  the  same  axis. 


CHAPTER  XVI. 
MOMENTS   OF  INERTIA  AND  PRINCIPAL   AXES. 

1.  Introduction. 

374.  As  will  be  shown  in  Chapters  XVII  and  XVIII,  the 
rotation  of  a  rigid  body  about  any  axis  depends  not  only  on 
the  forces  acting  on  the  body,  but  also  on  the  way  in  which 
the  mass  is  distributed  throughout  the  body.  This  distribu- 
tion of  mass  is  characterized  by  the  position  of  the  centroid 
and  by  that  of  certain  lines  in  the  body  called  principal  axes. 

It  has  been  shown  in  Art.  159  that  the  centroid  of  a  system 
of  particles  is  found  by  determining  the  moments,  or  more 
precisely,  the  moinents  of  the  first  order,  'Emx,  Zmy,  'Zmz,  of 
the  system  with  respect  to  the  co-ordinate  planes,  i.  e.  the 
sums  of  all  mass-particles  m  each  multipUed  l^y  its  distance 
from  the  co-ordinate  plane. 

The  principal  axes  of  a  sj'stem  of  particles  can  be  found  by 
determining  the  moments  of  the  second  order,  'Lmx'^,  '^my^, 
Iimz^,  llmyz,  Xmzx,  'Zmxy  of  the  system  with  respect  to  the 
same  planes.  We  proceed,  therefore,  to  study  the  theory  of 
such  moments. 

375.  If  in  a  rigid  body  the  mass  m  of  each  particle  be  multi- 
plied by  the  square  of  its  distance  r  from  a  given  point,  plane, 
or  line,  the  sum 

Zmr~  =  niiri~  +  nur2~  +  •  •  •  , 

extended  over  the  whole  body,  is  called  the  quadratic  moment, 
or,  more  commonly,  the  moment  of  inertia  of  the  body  for 
that  point,  plane,  or  line. 

280 


377.]     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     281 

If  the  body  is  not  composed  of  discrete  particles,  but  forms 
a  continuous  mass  of  one,  two,  or  three  dimensions,  this  mass 
can  be  resolved  into  elements  of  mass  dm,  and  the  sum  Smr^ 
becomes  a  single,  double,  or  triple  integral  J  r'^dm. 

Expressions  of  the  form  Zmvir^,  or  J  riVodm,  where  ri,  r2  are 
the  distances  of  m  or  of  dm  from  two  planes  (usually  at  right 
angles),  are  called  moments  of  deviation  or  products  of  inertia. 

376.  The  determination  of  the  moment  of  inertia  of  a  con- 
tinuous mass  is  a  mere  prol^lem  of  integration;  the  methods 
are  similar  to  those  for  finding  the  moments  of  mass  of  the 
first  order  required  for  determining  centroids,  the  only  dif- 
ference being  that  each  element  of  mass  must  be  multiplied 
by  the  square,  instead  of  the  first  power,  of  the  distance. 

A  moment  of  inertia  is  not  a  directed  c^uantity ;  it  is  not  a 
vector,  but  a  scalar;  indeed,  it  is  a  positive  ciuantity,  provided 
the  masses  are  all  positive,  as  we  shall  here  assume. 

If  the  mass  is  homogeneous,  the  density  appears  merely  as 
a  constant  factor;  as  the  density  in  this  case  can  be  regarded 
as  =1,  it  is  customary  to  speak  of  moments  of  inertia  of 
volumes,  areas,  and  lines. 

The  moment  of  inertia  of  any  number  of  bodies  ot  masses 
for  any  given  point,  plane,  or  line  is  obviously  the  sum  of  the 
moments  of  inertia  of  the  separate  bodies  or  masses  for  the 
same  point,  plane,  or  line. 

377.  The  moment  of  inertia  Xmr"^  of  any  body  whose  mass 
is  M  =  Sm  can  always  be  expressed  in  the  form 

2mr2  =  M-ro-, 

where  Vo  is  a  length  called  the  radius  of  inertia,  arm  of  inertia, 
or  radius  of  gyration.  This  length  ro  is  evidently  a  kind  of 
average  value  of  the  distances  r,  its  value  being  intermediate 
between  the  greatest  r'  and  least  r"  of  these  distances  r.     For 


282  KINETICS  [378. 

we  have  Swr'^  =  Smr^  =  'Lmr"'^,   or,    since    Swr'^  =  Mr'^, 
Xmr^  =  Mro\  ^mr"^  =  Mr'"", 

r'  ^  ro  =  r". 

378.  As  an  example,  let  us  determine  the  moment  of  inertia  of  a 
homogeneous  rectilinear  segment  (straight  rod  or  wire  of  constant  cross- 
section  and  density)  for  its  middle  point  (or  what  amounts  to  the  same 
thing,  for  a  line  or  plane  through  this  point  at  right  angles  to  the  seg- 
ment). 

Let  I  be  the  length  of  the  rod  (Fig.  80),  0  its  middle  point,  p"  its 

density  {i.  e.  the  mass  of  unit  length),  x  the  distance  OP  of  any  element 

J  dm  =  p"dx    from    the    middle 

Jt i +H ^       point.       Observing     that     the 

xp-     QQ  moment  of  inertia  for  O  of  the 

whole  rod  AB  \s  the  sum  of  the 

moments  of  inertia  of  the  halves  AO  and  OB,  and  that  the  moments 

of  inertia  of  these  halves  are  equal,  we  have,  for  the  moment  of  inertia 

/  of  AB, 

I  =  2£^x^.p"dx  =  ^p'% 
and  for  the  radius  of  inertia  ro,  since  the  whole  mass  is  M  =  p"  I, 

379.  Exercises. 

Determine  the  radius  of  inertia  in  the  following  cases.  When 
nothing  is  said  to  the  contrary,  the  masses  are  supposed  to  be  homo- 
geneous. 

(1)  Segment  of  straight  line  of  length  I,  for  perpendicular  through 
one  end. 

(2)  Rectangular  area  of  length  I  and  width  h:  (a)  for  the  side  h; 
(6)  for  the  side  I;  (c)  for  a  line  through  the  centroid  parallel  to  the 
side  h;  (d)  for  a  line  through  the  centroid  parallel  to  the  side  I. 

(3)  Triangular  area  of  base  b  and  height  h,  for  a  line  through  the 
vertex  parallel  to  the  base. 

(4)  Square  of  side  a,  for  a  diagonal. 

(5)  Regular  hexagon  of  side  a,  for  a  diagonal. 

(6)  Right  cylinder  or  prism  of  height  h,  for  the  plane  bisecting  the 
height  at  right  angles. 


381.]    MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     283 

(7)  Segment  of  straight  line  of  length  I,  for  one  end,  when  the  density 
is  proportional  to  the  nth  power  of  the  distance  from  this  end.  Deduce 
from  this:  (c)  the  result  of  Ex.  (1);  (6)  that  of  Ex.  (3);  (c)  the  radius 
of  inertia  of  a  homogeneous  pyramid  or  cone  (right  or  oblique)  of  height 
h,  for  a  plane  tlxrough  the  vertex  parallel  to  the  base. 

(8)  Circular  area  (plate,  disk,  lamina)  of  radius  a,  for  any  diameter. 

(9)  Circular  line  (wire)  of  radius  a,  for  a  diameter. 

(10)  Solid  sphere,  for  a  diametral  plane. 

(11)  Solid  ellipsoid,  for  the  three  principal  planes. 

(12)  Area  of  ring  bounded  by  concentric  circles  of  radii  ai,  aj,  for  a 
diameter. 

380.  The  moment  of  inertia  of  any  mass  AI  for  a  point  can 
easily  be  found  if  the  moments  of  inertia  of  the  same  mass 

are  known  for  any  hne  passing  through 
the  point,  and  for  the  plane  through  the 
I     point  perpendicular  to  the  line.     Let  0 
(Fig.  81)  be  the  point,  /  the  line,  tt  the 
plane;     r,  q,  p    the    perpendicular   dis- 
tances of  any  particle  of  mass  m  from 
0,  I,  T,    respectively.      Then  we  have, 
evidently,  r^  =  g^  +  p~.     Hence,   multi- 
Fig.  81.  plying  by  m,  and  summing  over  the 
whole  mass  M, 

^mr"^  =  Hmq^  +  '^mp'^;  (1) 

or,  putting  ^mr-  =  Mro~,  '^rnq'^  =  Mq^-,  1,mp^  =  Mpo"^,  where 
^0,  qo,  Po  are  the  radii  of  inertia  for  0,  I,  tt, 

To'  =  qo'  +  Po'.  (10 

381.  The  moment  of  inertia  of  any  mass  M  for  a  line  is 
equal  to  the  sum  of  the  moments  of  inertia  of  the  same  mass 
for  any  two  rectangular  planes  passing  through  the  line. 
Thus,  in  particular,  the  moment  of  inertia  for  the  axis  of  x  in 
a  rectangular  system  of  co-ordinates  is  equal  to  the  sum  of 


284 


KINETICS 


[382. 


the  moments  of  inertia  for  the  2a:-plane  and  x?/-plane.  This 
follows  at  once  by  considering  that  the  square  of  the  distance 
of  any  point  from  the  line  is  equal  to  the  sum  of  the  squares 
of  the  distances  of  the  same  point  from  the  two  planes.  Thus, 
if  q  be  the  distance  of  any  point  {x,  y,  z)  from  the  axis  of  x, 
we  have  g^  =  y"^  -\-  z-;  whence 

Zmq-  =  '^my-  -\-  Zmz"^. 

382.  It  follows,  from  the  last  article,  that  the  moment  of 
inertia  I^  of  a  plane  area,  for  any  line  perpendicular  to  its 
plane,  is 

if  ly,  Iz  are  the  moments  of  inertia  of  the  area  for  any  two 
rectangular  lines  in  the  plane  through 
the  foot  of  the  perpendicular  line. 

383.  The  problem  of  finding  the  mo- 
ment of  inertia  of  a  given  inass  for  a 
line  I',  when  it  is  ktiown  for  a  parallel 
line  I,  is  of  great  importance. 

Let  Smg^  be  the  moment  of  inertia 
of  the  given  mass  for  the  line  I  (Fig. 
82),  Smg'2  that  for  a  parallel  line  V  at 
the  distance  d  from  I.  The  distances 
q,  q'  of  any  particle  m  from  I,  V  form  with  d  a  triangle  which 
gives  the  relation 

g'2  =  g2  _|.  fp  _  2qd  cos{q,  d). 

Multiplying  by  m,  and  summing  over  the  whole  mass  M,  we 

find 

Zmq'~  =  2?ng2  +  Md^  -  2d1mq  cos(g,  d). 

Now  the  figure  shows  that  the  product  q  cos(q,  d)  in  the 
last  term  is  the  distance  p  of  the  particle  ?n  from  a  plane 


Fig.  S2. 


385.1     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     285 

through  I  at  right  angles  to  the  plane  determined  by  /  and  V. 
We  have,  therefore, 

Zmq'^  =  i:mq^  +  Aid'-  -  2d'Emp,  .  (2) 

where  the  last  term  contains  the  moment  of  the  first  order 

Zm/j  =  Mjj  of  the  given  mass  M  for  the  plane  just  mentioned. 

If,  in  particular,  this  plane  contains  the  centroid  G  of  the 

mass  M,  we  have  I,mp  =  0,  so  that  the  formula  reduces  to 

2mg'2  =  ^mq^  -j.  Md\  (3) 

Introducing  the  radii  of  inertia  50',  Qo,  this  can  be  written 

go'-  =  go-  +  d'\  (3') 

384.  Similar  considerations  hold  for  the  moments  of  inertia 
Zmp"^,  Zmp'^  with  respect  to  two  parallel  planes  tt,  t'  at  the 
distance  d  from  each  other.  We  have,  in  this  case,  p'  = 
p  —  d;  hence, 

2wp'2  =  v^p2  _|_  ]^f^2  _  2cZ2mp,  (4) 

and  if  the  plane  x  contain  the  centroid  G, 

385.  Of  special  importance  is  the  case  in  which  one  of  the 
lines  (or  planes),  say  Z(x),  contains  the  centroid.  The  for- 
mulae (3),  (3'),  and  (5)  hold  in  this  case;  and  if  we  agree  to 
designate  any  line  (plane)  passing  through  the  centroid  as  a 
centroidal  line  (plane),  our  proposition  can  be  expressed  as 
follows :  The  moment  of  inertia  for  any  line  (plane)  is  found 
from  the  moment  of  inertia  for  the  parallel  centroidal  line  {plane) 
by  adding  to  the  latter  the  product  Md^  of  the  whole  mass  into 
the  square  of  the  distance  of  the  lines  (planes). 

It  will  be  noticed  that  of  all  parallel  fines  (planes)  the 
centroidal  line  (plane)  has  the  least  moment  of  inertia. 


286  KINETICS  [386. 

386.  Exercises. 

Determine  the  radius  of  inertia  of  the  following  homogeneous  masses : 

(1)  Rectangular  plate  of  length  I  and  width  h,  for  a  centroidal  line 
perpendicular  to  its  plane. 

(2)  Area  of  equilateral  triangle  of  side  a:  (a)  for  a  centroidal  line 
parallel  to  the  base;  (b)  for  an  altitude;  (c)  for  a  centroidal  line  per- 
pendicular to  its  plane. 

(3)  Circular  disk  of  radius  a:  (a)  for  a  tangent;  (6)  for  a  line  through 
the  center  perpendicular  to  the  plane  of  the  disk ;  (c)  for  a  perpendicular 
to  its  plane  through  a  point  in  the  circumference. 

(4)  Solid  sphere,  for  a  diameter. 

(5)  Area  of  ring  bounded  by  concentric  circles  of  radii  ai,  02,  for  a 
line  through  the  center  perpendicular  to  the  plane  of  the  ring. 

(6)  Right  circular  cylinder,  of  radius  a  and  height  h:  (a)  for  its 
axis;  (6)  for  a  generating  line;  (c)  for  a  centroidal  line  in  the  middle 
cross-section. 

(7)  By  Ex.  (3)  (6),  the  moment  of  inertia  of  the  area  of  a  circle  of 
radius  a,  for  its  axis  {i.  e.  the  perpendicular  to  its  plane,  passing  through 
the  center),  is  /  =  ^iraK     Differentiating  with  respect  to  a,  we  find: 

-7-  =  2-ira^  =  2ira  ■  a? ; 
da 

hence,  approximately  for  small  Aa: 

A7  =  27ra'Ao  =  2iraAa  •  a^. 

This  is  the  moment  of  inertia  of  the  thin  ring,  of  thickness  Ao,  for  its 
axis.     (Comp.  Ex.  (5).) 

If  the  constant  surface  density  (Art.  155)  of  the  circle  be  p',  we  have 
/  =  \p'Tra*\  hence 

A/  =  27rap'Aa  •  a^, 

where  p'Aa  is  the  linear  density  p"  of  the  ring. 

(8)  Apply  the  method  of  Ex.  (7)  to  derive  from  Ex.  (4)  the  moment 
of  inertia  of  a  thin  spherical  shell,  of  radius  a  and  thickness  Ao,  for  a 
diameter. 

(9)  Area  of  ellipse:  (a)  for  the  major  axis;  (6)  for  the  minor  axis; 
(c)  for  the  perpendicular  to  its  plane  through  the  center. 

(10)  Solid  ellipsoid,  for  each  of  the  three  axes. 

(11)  Wire  bent  into  an  equilateral  triangle  of  side  a,  for  a  centroidal 
line  at  right  angles  to  the  plane  of  the  triangle. 


388.]     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     287 

(12)  Paraboloid  of  revolution,  bounded  by  the  plane  through  the 
focus  at  right  angles  to  the  axis,  for  the  axis. 

(13)  Anchor-ring,  produced  by  the  revolution  of  a  circle  of  radius  a 
about  a  line  in  its  plane  at  the  distance  h{>  a)  from  the  center,  for  the 
axis  of  revolution. 

2.  Ellipsoids  of  inertia. 

387.  The  moments  of  inertia  of  a  given  mass  for  the  dif- 
ferent hnes  of  space  are  not  independent  of  each  other. 
Several  examples  of  this  have  already  been  given.  It  has 
been  shown,  in  particular  (Art.  383),  that  if  the  moment  of 
inertia  be  knov/n  for  any  line,  it  can  be  found  for  any  parallel 
line.  It  follows  that  if  the  moments  be  known  for  all  lines 
through  any  given  point,  the  moments  for  all  lines  of  space 
can  be  found.  We  now  proceed  to  study  the  relations  be- 
tween the  moments  of  inertia  for  all  the  lines  passing  through 
any  given  point  0. 

Let  X,  y,  z  be  the  co-ordinates  of  any  particle  m  of  the  mass; 
and  let  us  denote  hy  A,  B,  C  the  moments  of  inertia  of  M  for 
the  axes  of  x,  y,  z;  by  A',  B',  C  those  for  the  planes  ijz,  zx,  xy; 
by  D,  E,  F  the  products  of  inertia  (Art.  375)  for  the  co- 
ordinate planes;  i.  e.  let  us  put 

A  =  Zm(?/2  +  z^),         A'  =  2mx%         D  =  2myz, 

B  =  -Emiz^  +  a;2),         B'  =  l^imf,         E  =  Zmzx,      (6) 

C  =  2m(a;2  +  ij^),         C  =  Sms^,  F  =  ^mxy. 

388.  These  nine  quantities  are  not  independent  of  each 
other.     We  have  evidently 

A  =  B'  +  C\     B  =  C  +  A',     C  =  A'  +  B'; 

hence,  solving  for  A',  B' ,  C, 

A'^UB-\-C-A),    B'=UC+A-B),   C'=^A-^B-C). 
The  moment  of  inertia  for  the  origin  0  is 

Swr2=Sm(a;2+ 1/2+22)  =  A'-j-  B'-\- C  =  i{A -\- B-\-C).     (7) 


288 


KINETICS 


[389. 


389.  The  moment  of  inertia  I  for  any  line  through  0  can 
be  expressed  by  means  of  the  six  quantities  A,  B,C,  D,  E,  F; 
and  the  moment  of  inertia  /'  for  any  plane  through  0  can  be 
expressed  by  means  of  A',  B',  C,  D,  E,  F. 

Let  TT  (Fig.  83)  be  any  plane  passing  through  0;  Z  its  nor- 
mal; a,  /3,  7  the  direction  cosines  of  I;  and,  as  before  (Art. 

380),  p,  q,  r  the  distances  of 
any  point  {x,  y,  z)  of  the  given 
mass  from  tt,  I,  and  0,  re- 
spectively. Then,  projecting 
the  closed  polygon  formed  by 
r,  X,  \j,  z  on  the  line  I,  we  have 

p  =  ax  +  /3?/  +  72; 


hence,   squaring,    multiplying 
by  w,  and  summing  over  the 


Fig.  83. 
whole  mass,  we  find 

+  2l3y'^myz  +  2yaZmzx  +  2a^'Emxy, 
or,  with  the  notations  (6), 

r  =  A' a'-  +  B'I3-  +  CY-  +  2Z)/37  +  2jE'7a  +  2Fa/3.     (8) 

Thus  the  moment  of  inertia  for  any  j)lane  through  the  origin 
is  expressed  as  a  homogeneous  quadratic  function  of  the  direction 
cosines  of  the  normal  of  the  plane. 

390.  The  moment  of  inertia  I  =  llinq^  for  the  line  I  can 
now  be  found  from  equation  (1),  Art.  380,  by  substituting  for 
I,7nr'^  and  Zmj)^  their  values  from  (7)  and  (8) : 

I  =  i:mr^  -  7'  -  A'  +  B'  +  C  -  I' 

=  A'(l-a'-)-\-B'{l-(3'-)-\rC'{l-y'-)-2D^y-2Eya-2Fa^, 

or,  since  a-  +  /3-  +  7-  =  1, 


392.]     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     289 

I  =  A'(^'  +  7')  +  B'W  +  a^)  +  C'(a2  +  iS^) 

=  a'-iB'  +  C)  +  fi\C'  +  A')  +  7-(A'  +  5') 

-  2D^y  -  2Eya  -  2Fa^; 

hence,  finally,  applying  the  relations  of  Art.  388, 

I  =  Aa''^  5/32  +  (7^2  _  27)^^  _  2Eya  -  2F«/3.       (9) 

The  moment  of  inertia  for  amj  line  through  the  origin  is, 
therefore,  also  a  homoge^ieous  quadratic  function  of  the  direction 
cosines  of  the  line. 

391.  These  results  suggest  a  geometrical  interpretation. 
Imagine  an  arbitrary  length  OP  =  p  laid  off  from  the  origin 
0  on  the  line  I  whose  direction  cosines  are  a,  /3,  7;  the  co- 
ordinates of  the  extremity  P  of  this  length  will  be  a:  =  pa, 
y  =  p^,  z  =  py.  Now,  if  equation  (9)  be  multiplied  by  p^, 
it  assumes  the  form 

Ax^  +  By^  +  Cz^  -  2Dyz  -  2Ezx  -  2Fxy  =  pU, 

which  represents  a  quadric  surface  provided  that  p  be  selected 
for  the  different  lines  through  0  so  as  to  make  p^/  constant, 
say  p2/  =  K^.  Hence,?/  on  every  line  I  through  the  origin  a 
length  OP  =  p  =  kJ  -^  I  he  laid  off,  i.  e.  a  length  inversely  pro- 
portional to  the  square  root  of  the  moment  of  inertia  I  for  this 
line  I,  the  points  P  will  lie  on  the  quadric  surface 

Ax^  +  By^  +  C22  -  2Dyz  -  2Ezx  -  2Fxy  =  k\ 

The  constant  k^  may  be  selected  arbitrarily;  to  preserve 
the  homogeneity  of  the  equation  it  will  l^e  convenient  to 
take  it  in  the  form  k^  =  Me*,  where  e  is  an  arl)itrary  length. 

392.  As  moments  of  inertia  are  essentially  positive  quan- 
tities, the  radii  vectores  of  the  surface 

Ax^  +  By^-  +  C22  -  2Dyz  -  2Ezx  -  2Fxy  =  Me*    (10) 
20 


290  KINETICS  [393. 

are  all  real,  and  the  surface  is  an  ellipsoid.  It  is  called  the 
ellipsoid  of  inertia,  or  the  momental  ellipsoid,  of  the  point  0. 
This  point  0  is  the  center;  the  axes  of  the  ellipsoid  are  called 
the  principal  axes  at  the  point  0;  and  the  moments  of  inertia 
for  these  axes  are  called  the  principal  7noments  of  inertia  at  the 
point  0.  Among  these  there  will  evidently  be  the  greatest 
and  least  of  all  the  moments  of  the  point  0,  the  greatest 
moment  corresponding  to  the  shortest,  the  least  to  the  longest 
axis  of  the  ellipsoid. 

It  may  be  observed  that,  owing  to  the  relations  of  Art.  388, 
which  show  that  the  sum  of  any  two  of  the  quantities  A,  B,C 
is  always  greater  than  the  third,  not  every  ellipsoid  can  be 
regarded  as  the  momental  ellipsoid  of  some  mass.  An  ellip- 
soid can  be  a  momental  ellipsoid  only  when  a  triangle  can  be 
constructed  of  the  reciprocals  of  the  squares  of  its  semi-axes. 

393.  If  the  axes  of  the  ellipsoid  (10)  be  taken  as  axes  of 
co-ordinates,  the  equation  assumes  the  form 

hx^  +  hy-  +  hz-  =  Me\  (11) 

where  7i,  h,  I3  are  the  principal  moments  at  the  point  0. 

By  Art.  391  we  have  p^  =  k'-/I  =  Me\fl;  hence  7  =  Me*/p\ 
If,  therefore,  equation  (11)  be  divided  by  p-,  the  following 
simple  expression  is  obtained  for  finding  the  moment  of 
inertia,  I,  for  a  line  whose  direction  cosines  referred  to  the 
principal  axes  are  a,  /3,  7: 

I  =  /la^  +  W  +  hy^-  (12) 

394.  To  make  use  of  this  form  for  7,  the  principal  axes  at  the  point 
0,  i.  e.  the  axes  of  the  momental  ellipsoid  (10),  must  be  known.  The 
determination  of  the  axes  of  an  ellipsoid  whose  equation  referred  to 
the  center  is  given  is  a  well-known  problem  of  analytic  geometry.  It  can 
be  solved  by  considering  that  the  semi-axes  are  those  radii  vectores  of 
the  surface  that  are  normal  to  it.     The  direction  cosines  of  the  normal 


394.1     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     291 

of  any  surface  F{x,  y,  z)  =0  are  proportional  to  the  partial  derivatives 
dFjdx,  dF/dy,  dF/dz.  If,  therefore,  the  radius  vector  p  is  a  semi-axis, 
its  direction-cosines  a,  /3,  y  must  be  proportional  to  the  partial  deriv- 
atives of  the  left-hand  member  of  (10);  i.  e.  we  must  have 

Ax  -  Fy  -  Ez  ^-  Fx  +  By  -  Dz  ^  -  Ex  -  Dy  +  Cz 
a  "  fi  7  ' 

or  dividing  the  numerators  by  p, 

Aa  -  F0  -  Ey  ^  -  Fa  +  B0  -  Dy  ^  -  Ea  -  D0  +  Cy 
a  ^  y  ' 

Denoting  the  common  value  of  these  fractions  by  /  we  have 
al  =  Aa  -  Fp  -  Ey,    /3/  =  -  Fa  +  Bfi  -  Dy,    yl  =  -  Ea  -  Dff+Cy] 
multiplying  these  equations  by  a,  p,  y  and  adding  we  find 

I  =  Aa'^  +  Bff^  +  Cy^  -  2D^y  -  2Eya  -  2Fa/3, 

which,  compared  with  (9),  shows  that  /  is  the  moment  of  inertia  for 
the  axis  (a,  /3,  y).  To  obtain  it  in  terms  oi  A,B,  C,  D,  E,  F,  we  write 
the  preceding  three  equations  in  the  form 

(/  -  A)a  +  F^  +  Ey  =0, 

Fa  +  {I  -  B)p  +  Dy  =0,  (13) 

Ea+  D^+  {I  -  C)y  =  0, 

whence,  eliminating  a,  /3,  y,  we  find  /  determined  by  the  cubic  equation 

I  -  A,  F,  E 

F,I  -  B,  D 

E,         D,I  -C 


=  0.  (14) 


The  roots  of  this  cubic  are  the  three  principal  moments  I\,  1 2,  h  of 
the  point  0.  The  direction  cosines  of  the  principal  axes  are  then  found 
by  substituting  successively  I\,  h,  h  in  (13)  and  solving  for  a,  /8,  y. 

In  general,  the  three  principal  moments  of  inertia  I],  h,  h  at  a  point 
O  are  different.  If,  however,  two  of  them  are  equal,  saj'  h  =  h,  the 
momental  ellipsoid  becomes  an  ellipsoid  of  revolution  about  the  third, 
/i,  as  axis;  and  it  follows  that  the  moments  of  inertia  for  all  lines 
through  O  in  the  plane  of  the  two  equal  axes  are  equal. 

If  I\  =  h  =  I3,  the  ellipsoid  becomes  a  sphere,  and  the  moments  of 
inertia  are  the  same  for  all  lines  passing  through  0. 


292  KINETICS  [395. 

395.  If  the  equation  of  the  momental  elhpsoid  at  a  point  0  be  of  the 
form   Ax''  +  B^f  +  Cz^  —  2Dyz  =  Ale*,    i.    e.    if    the   two    conditions 

E  =  Xmzx  =0,     F  =  i:mxy  =  0 

be  fulfilled,  the  axis  of  x  coincides  with  one  of  the  three  axes  of  the 
ellipsoid,  the  surface  being  symmetrical  with  respect  to  the  yz-plane. 
Hence,  if  the  coriditions  E  =  0,  F  =  0  are  satisfied,  the  axis  of  x  is  a 
'principal  axis  at  the  origin.  The  converse  is  evidently  also  true;  i.  e. 
if  a  line  is  a  principal  axis  at  one  of  its  points,  then,  for  this  point  as 
origin  and  the  line  as  axis  of  x,  the  conditions  'Zmzx  =  0,  I^mxy  =  0 
must  be  satisfied. 

It  is  easy  to  see  that  if  a  Une  be  a  principal  axis  at  one  of  its  points, 
say  0,  it  will  in  general  not  be  a  principal  axis  at  any  other  one  of  its 
points.  For,  taking  the  line  as  axis  of  x  and  0  as  origin,  we  have 
Hjtizx  =  0,  '^mxy  =  0.  If  now  for  a  point  0'  on  this  line  at  the  dis- 
tance a  from  0  the  line  is  likewise  a  principal  axis,  the  conditions 

2??iz(x  —  a)  =  0,     Zm{x  —  a)y  =  0 

must  be  fulfilled.     These  reduce  to 

2m2  =  0,     "Liny  =  0, 

and  show  that  the  line  must  pass  through  the  centroid.  And  as  for  a 
centroidal  line  these  conditions  are  satisfied  independently  of  the  value 
of  a,  it  appears  that  a  centroidal  principal  axis  is  a  principal  axis  at  every 
one  of  its  points.  Hence,  a  line  cannot  be  principal  axis  at  more  than 
one  of  its  points  unless  it  pass  through  the  centroid;  in  the  latter  case  it  is 
a  principal  axis  at  every  one  of  its  points. 

396.  All  those  lines  passing  through  a  given  point  0  for  which  the 
moments  of  inertia  have  the  same  value  I  can  be  shown  to  form  a  cone 
of  the  second  order  whose  principal  diameters  coincide  with  the  axes  of 
the  momental  ellipsoid  at  0.  This  is  called  an  equimomental  cone. 
Its  equation  is  obtained  by  regarding  /  as  constant  in  equation  (12) 
and  introducing  rectangular  co-ordinates.  Multiplying  (12)  by  a"^  + 
/3^  +  7^  =  1,  we  find 

(7i  -  IW  +  ih  -  1)0'  +  (h  -  IW  =  0; 

and  multiplying  by  p-,  we  obtain  the  equation  of  the  equimomental  cone 
in  the  form 

(7i  -  I)x'  +  {h  -  I)y-  +  (/3  -  7)2^  =  0.  (15) 


397.]     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     293 

397.  A  slightly  different  form  of  the  equations  (11),  (12),  (15)  is 
often  more  convenient ;  it  is  obtained  by  introducing  the  three  principal 
radii  of  inertia  gi,  q^,  qz  defined  by  the  relations 

/i  =  Mgi2,     h  =  Mq2^,     h  =  Mq^"^. 

The  equation  (11)  of  the  momental  ellipsoid  at  the  point  0  then  as- 
sumes the  form 

qi^x^  +  92^  +  gaV  =  64.  (11') 

The  expression  of  the  radius  of  inertia  q  for  any  line  (a,  /3,  y)  through 
0  becomes 

g2    =    ^^2^2   +  g,2^2  +  532^2.  (12') 

Dividing  (11')  by  the  square  of  the  radius  vector,  p^,  and  comparing 
with  (12'),  we  find 

q  =  ~  ,     p  =  -~,  (16) 

P  9 

as  is  otherwise  apparent  from  the  fundamental  property  of  the  momen- 
tal ellipsoid  (Art.  391). 

The  equation  of  the  equimomental  cone  for  all  whose  generators  the 
radius  of  inertia  has  the  value  q  is  obtained  from  (15)  in  the  form 

(5,2  _  52)3.2  +  (5^2  _  g2)y  +  (532  _  ^2)^2  =  0.  (15') 

This  cone  meets  any  one  of  the  momental  ellipsoids  (11')  in  points 
whose  radii  vectores  p  are  all  equal;  and  if  we  select  the  arbitrary  con- 
stant e  equal  to  the  radius  of  inertia  q  of  the  generators  of  the  equi- 
momental cone,  it  follows  from  (16)  that  p  =  q.  This  particular 
ellipsoid  has  the  equation 

7i-x2  ^  5,2,^2  _|_  532^2  =  qi^ 

and  its  intersection  with  the  equimomental  cone  (15')  lies  on  the  sphere 
a;2  _[_  2/2  _|_  22  =  qi^ 

In  other  words,  the  equimomental  cone  (15')  passes  through  the  sphero- 
conic  in  which  the  ellipsoid  meets  the  sphere.  Multiplying  the  equa- 
tion of  the  sphere  by  q^  and  subtracting  it 'from  the  equation  of  the 
ellipsoid  wc  obtain  the  equation  (15')  of  the  cone. 

The  semi-axes  of  the  ellipsoid  are  q^/qi,  q-jq^,  q^/qs.  If  we  assume 
h  >  h  >  h,  and  hence  qi  >  qi  >  qz,  q  must  be  ^q^/qs  and  =5V<Zi 
As  long  as  q  is  less  than  the  middle  somi-axis  q'^/qi  of  the  ellipsoid,  the 


294 


KINETICS 


[398. 


axis  of  the  cone  coincides  with  the  axis  of  x;  but  when  q  >  (fjqi,  the 
axis  of  2  is  the  axis  of  the  cone.  For  q  =  q^jq^,  i.  e.  q  =  q2,  the  cone 
(15')  degenerates  into  the  pair  of  planes  (gi^  —  q2^)x'^  —  {q^^  —  qz^)z^  =  0. 
These  are  the  planes  of  the  central  circular  (or  cyclic)  sections  of  the 
elUpsoid;  they  divide  the  elUpsoid  into  four  wedges,  of  which  one  pair 
contains  all  the  equimomental  cones  whose  axes  coincide  with  the 
greatest  axis  of  the  ellipsoid,  while  the  other  pair  contains  all  those 
whose  axes  lie  along  the  least  axis  of  the  ellipsoid. 

398.  There  is  another  ellipsoid  closely  connected  with  the  theory 
of  principal  axes;  it  is  obtained  from  the  momental  elUpsoid  by  the 
process  of  reciprocation. 

About  any  point  0  (Fig.  84)  as  center  let  us  describe  a  sphere 
of  radius  e,  and  construct  for  every  point  P  its  polar  plane  tt  with 

regard  to  the  sphere.  If  P  describe  any 
surface,  the  plane  -k  will  envelop  another 
surface  which  is  called  the  -polar  reciprocal 
of  the  former  surface  with  regard  to  the 
sphere. 

Let  Q  be  the  intersection  of  OP 
with  TT,  and  put  OP  =  p,  OQ  =  q;  then 
it  appears  from  the  figure  that 


pq  =  t' 


(16) 


Fig.  84. 
the  ellipsoid 


399.  It  is  easy  to  see  that  the  polar 
reciprocal  of  the  momental  ellipsoid  (11') 
with  respect  to  the  sphere  of  radius  e  is 


+ 


q^^ 


1. 


(17) 


To  prove  this  it  is  only  necessary  to  show  that  the  relation  (16)  is 
fulfilled  for  p  as  radius  vector  of  (11'),  and  q  as  perpendicular  to  the 
tangent  plane  of  (17).     Now  this  tangent  plane  has  the  equation 


4x  +  -^7  +  ;f,-Z  =  l; 
q-^  q^         qr 

hence  we  have  for  the  direction  cosines  a,  /3,  y  and  for  the  length  q 
of  the  perpendicular  to  the  tangent  plane 


401.]     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     295 


7 


xlqi^         ylqi^         z/q^'  [x'^/q.i  +  y'i/q^t  +  22/^34]^' 

These  relations  give  qia  =  {x/qi)q,  q^^  =  {ylqi)q,  qzy  =  {z/q3)q,  whence 

qM'  +  92^/32  +  532^2  =  C  ^  +  ^  +  ^\  g2  =  g2.  (18) 

\  qr       q-r       q^  J 

For  the  radius  vector  pof  (11')  whose  direction  cosines  a,  /3,  7  are  the 
same  as  those  of  q,  we  have  by  (11')- 


q-^oi^  +  52-/32  _|_  ^32^2 


Hence  p^(^  =  e^;  and  this  is  what  we  wished  to  prove. 

400.  The  surface  (17)  has  variously  been  called  the  ellipsoid  of  gyra- 
tion, the  ellipsoid  of  inertia,  the  reciprocal  ellipsoid.  We  shall  adopt 
the  last  name.  The  semi-ax.  s  of  this  ellipsoid  are  equal  to  the  princi- 
pal radii  of  inertia  at  the  point  0.  The  directions  of  its  axes  coincide 
with. those  of  the  momental  ellipsoid;  but  the  greatest  axis  of  the  former 
coincides  with  the  least  of  the  latter,  and  vice  versa. 

By  comparing  the  equations  (12')  and  (IS)  it  will  be  seen  that  q  is 
the  radius  of  inertia  of  the  line  {a,  0,  7)  on  which  it  Ues.  Thus,  while 
the  radius  vector  OP  =  p  of  the  momental  ellipsoid  is  inversely  propor- 
tional to  the  radius  of  inertia,  i.  e.  p  —  e-/q,  the  reciprocal  ellipsoid  gives 
the  radius  of  inertia  q  for  a  line  as  the  segment  cut  off  on  this  line  by  the 
perpendicrdar  tangent  plane. 

401.  We  are  now  prepared  to  determine  the  moment  of  inertia  for 
any  line  in  space.  Let  us  construct  at  the  centroid  G  of  the  given 
mass  or  body  both  the  momental  ellipsoid  and  its  polar  reciprocal. 
The  former  is  usually  called  the  central  ellipsoid  of  the  body;  the  latter 
we  may  call  the  fundamental  ellipsoid  of  the  body.  As  soon  as  this 
fundamental  ellipsoid 

^  -4_  '/   j_  ii  =  1 

51'     q2'     33^ 

is  known,  the  moment  of  inertia  of  the  body  for  any  line  whatever  can 
readily  be  found.  For,  by  Art.  400,  the  radius  of  inertia  q  for  any  line 
lo  passing  through  the  centroid  is  equal  to  the  segment  OQ  cut  off  on 
the  line  Iq  by  the  perpendicular  tangent  plane  of  the  fundamental 
ellipsoid;  and  for  any  line  I  not  passing  through  the  centroid,  the 
square  of  the  radius  of  inertia  can  be  deterininod  by  first  finding  the 


296  KINETICS  [402. 

square  of  the  radius  of  inertia  for  the  parallel  centroidal  line  k,  and 
then,  by  Art.  385,  adding  to  it  the  square  of  the  distance  d  of  the 
centroid  from  the  line  I. 

402.  In  the  problem  of  determining  the  ellipsoids  of  inertia  for  a 
given  body  at  any  point,  considerations  of  symmetry  are  of  great 
assistance. 

Suppose  a  given  mass  to  have  a  plane  of  symmetry;  then  taking 
this  plane  as  the  ?/z-plane,  and  a  perpendicular  to  it  as  the  axis  of 
X,  there  must  be,  for  every  particle  of  mass  m,  whose  co-ordinates  are 
X,  y,  z,  another  particle  of  equal  mass  m,  whose  co-ordinates  are  —  x,  y, 
z.  It  follows  that  the  two  products  of  inertia  Smzx  and  Smxi/  both 
vanish,  whatever  the  position  of  the  other  two  co-ordinate  planes. 
Hence,  any  perpendicular  to  the  plane  of  symmetry  is  a  principal  axis 
at  its  point  of  intersection  with  this  plane. 

Let  the  mass  have  two  planes  of  symmetry  at  right  angles  to  each 
other;  then  taking  one  as  ?/z-plane,  the  other  as  zx-plane,  and  hence 
their  intersection  as  axis  of  x,  it  is  evident  that  all  three  products  of 
inertia  vanish, 

S?n?/z  =  0,         'Lmzx  =  0,         'Zmxy  =  0, 

wherever  the  origin  be  taken  on  the  intersection  of  the  two  planes. 
Hence,  for  any  point  on  this  intersection,  the  principal  axes  are  the 
line  of  intersection  of  the  two  planes  of  symmetry,  and  the  two  per- 
pendiculars to  it,  drawn  in  each  plane. 

If  there  be  three  planes  of  symmetry,  their  point  of  intersection 
is  the  centroid,  and  their  lines  of  intersection  are  the  principal  axes 
at  the  centroid. 

403.  Exercises. 

Determine  the  principal  axes  and  radii  at  the  centroid,  the  central 
and  fundamental  ellipsoids,  and  show  how  to  find  the  moment  of  inertia 
for  any  line,  in  the  following  Exercises  (1),  (2),  (3). 

(1)  Rectangular  parallelepiped,  the  edges  being  2a,  26,  2c.  Find 
also  the  moments  of  inertia  for  the  edges  and  diagonals,  and  specialize 
for  the  cube. 

(2)  Ellipsoid  of  semi-axes  a,  b,  c.  Determine  also  the  radius  of 
inertia  for  a  parallel  I  to  the  shortest  axis  passing  through  the  extremity 
of  the  longest  axis. 

(3.)  Right  circular  cone  of  height  h  and  radius  of  base  a.     Find 


404.]     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     297 

first  the  principal  moments  at  the  vertex;  then  transfer  to  the  centroid. 

(4)  Determine  the  momental  eUipsoid  and  the  principal  axes  at  a 
vertex  of  a  cube  whose  edge  is  a. 

(5)  Determine  the  radius  of  inertia  of  a  tliin  wire  bent  into  a  circle, 
for  a  line  through  the  center  incUned  at  an  angle  a  to  the  plane  of  the 
circle. 

(6)  A  peg-top  is  composed  of  a  cone  of  height  H  and  radius  a,  and 
a  hemispherical  cap  of  the  same  radius.  The  pointed  end,  to  a  distance 
h  from  the  vertex  of  the  cone,  is  made  of  a  material  three  times  as  heavy 
as  the  rest.  Find  the  moment  of  inertia  for  the  axis  of  rotation; 
specialize  for  h  =  a  =  \H. 

(7)  Show  that  the  principal  axes  at  any  point  P,  situated  on  one  of 
the  principal  axes  of  a  body,  are  parallel  to  the  centroidal  principal  axes, 
and  find  their  moments  of  inertia. 

(8)  For  a  given  body  of  mass  M  find  the  points  {spherical  -points  of 
inertia)  at  which  the  momental  ellipsoid  reduces  to  a  sphere. 

(9)  Determine  a  homogeneous  ellipsoid  having  the  same  mass  as  a 
given  body,  and  such  that  its  moment  of  inertia  for  every  line  shall  be 
the  same  as  that  of  the  given  body. 

(10)  For  a  given  body  M,  whose  centroidal  principal  radii  are  qi,  qi, 
qs,  determine  three  homogeneous  straight  rods  intersecting  at  right 
angles,  of  such  lengths  2a,  26,  2c,  and  such  linear  density  p",  that  they 
have  the  same  mass  and  the  same  moment  of  inertia  (for  any  line)  as 
the  given  body. 

3.  Distribution  of  principal  axes  in  space. 

404.  It  has  been  shown  in  the  preceding  articles  how  the  principal 
axes  can  be  determined  at  any  particular  point.  The  distribution  of 
the  principal  axes  throughout  space  and  their  position  at  the  different 
points  is  brought  out  very  graphically  by  means  of  the  theory  of  con- 
focal  quadrics.  It  can  be  shown  that  the  directions  of  the  principal 
axes  at  any  point  are  those  of  the  principal  diameters  of  the  tangent 
cone  drawn  from  this  point  as  vertex  to  the  fundamental  ellipsoid;  or, 
what  amounts  to  the  same  thing,  thoy  are  the  directions  of  the  normals 
of  the  three  quadric  surfaces  passing  through  the  point  and  confocal 
to  the  fundamental  ellipsoid. 

In  order  to  explain  and  prove  these  propositions  it  will  be  necessary 
to  give  a  short  sketch  of  the  theory  of  confocal  conies  and  quadrics. 


298  KINETICS  [405. 

405.  Two  conic  sections  are  said  to  be  confocal  when  they  have  the 
same  foci.  The  directions  of  the  axes  of  all  conies  having  the  same 
two  points  S,  S'  as  foci  must  evidently  coincide,  and  the  equation  of 
such  conies  can  be  written  in  the  form 

where  X  is  an  arbitrary  parameter.  For,  whatever  value  may  be  as- 
signed in  this  equation  to  X,  the  distance  of  the  center  0  from  either 
focus  will  always  be  i^a^  +  X  -  (6^  +  X)  =  Va^  -  ¥;  it  is  therefore 
constant. 

406.  The  individual  curves  of  the  whole  system  of  confocal  conies 
represented  by  (19)  are  obtained  by  giving  to  X  any  particular  value 
between  —  oo  and  +  oo;  thus  we  may  speak  of  the  conic  X  of  the 
system. 

For  X  =  0  we  have  the  so-called  fundamental  conic  x^/a^  +  y'^/b-  =  1 ; 
this  is  an  ellipse.  To  fix  the  ideas  let  us  assume  a  >  b.  For  all  values 
of  X  >  —  b^,  i.  e.  as  long  as  —  6^  <  x  <  oo,  the  conies  (19)  are  ellipses, 
beginning  with  the  rectilinear  segment  SS'  (which  may  be  regarded  as 
a  degenerated  ellipse  X  =  —  6^  whose  minor  axis  is  0),  expanding  gradu- 
ally, passing  through  the  fundamental  elhpse  X  =  0,  and  finally  verging 
into  a  circle  of  infinite  radius  for  X  =  oo. 

It  is  thus  geometrically  evident  that  through  every  point  in  the  plane 
will  pass  one,  and  only  one,  of  these  ellipses. 

407.  Let  us  next  consider  what  the  equation  (19)  represents  when  X 
is  algebraically  less  than  —  6^.  The  values  of  X  that  are  <  —  a^  give 
imaginary  curves,  and  are  of  no  importance  for  our  purpose.  But  as 
long  as  —  a^  <  X  <  —  6^,  the  curves  are  hyperbolas.  The  curve  X  = 
—  b^  may  now  be  regarded  as  a  degenerated  hj^perbola  collapsed  into  the 
two  rays  issuing  in  opposite  directions  from  S  and  S'  along  the  line  SS'. 
The  degenerated  ellipse  together  with  this  degenerated  hyperbola  thus 
represents  the  whole  axis  of  x. 

As  X  decreases,  the  hyperbola  expands,  and  finally,  for  X  =  —  a^, 
verges  into  the  axis  of  y,  which  may  be  regarded  as  another  degenerated 
hj'perbola. 

The  system  of  confocal  hyperbolas  is  thus  seen  to  cover  likewise  the 
whole  plane  so  that  one,  and  only  one,  hjTjerbola  of  the  system  passes 
through  every  point  of  the  plane. 


409.]     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     299 

408.  The  fact  that  every  point  of  the  plane  has  one  ellipse  and  one 
hyperbola  of  the  confocal  system  (19)  passing  through  it,  enables  us  to 
regard  the  two  values  of  the  parameter  X  that  determine  these  two 
curves  as  co-ordinates  of  the  point;  they  are  called  elliptic  co-ordinates. 
If  X,  y  be  the  rectangular  cartesian  co-ordinates  of  the  point,  its  elliptic 
co-ordinates  Xi,  X2  are  found  as  the  roots  of  the  equation  (19)  which 
is  quadratic  in  X.  Conversely,  to  transform  from  elliptic  to  cartesian 
co-ordinates,  that  is,  to  express  x  and  y  in  terms  of  Xi  and  X2,  we  have 
only  to  solve  for  x  and  y  the  two  equations 

"^^     _L     y"^      =  1  3:^      I      y'^      ^  ■, 


a2  +  Xi      62  ^-  Xi         '     02^X26^  +  X2 

The  two  confocal  conies  that  pass  through  the  same  point  P  intersect 
at  right  angles.  For,  the  tangent  to  the  ellipse  at  P  bisects  the  exterior 
angle  at  P  in  the  triangle  SPS',  while  the  tangent  to  the  hyperbola 
bisects  the  interior  angle  at  the  same  point;  in  other  words,  the  tangent 
to  one  curve  is  normal  to  the  other,  and  vice  versa.  The  elliptic  system 
of  co-ordinates  is,  therefore,  an  orthogonal  system;  the  infinitesimal 
elements  dXi  ■  dX2  into  which  the  two  series  of  confocal  conies  (19) 
divide  the  plane  are  rectangular,  though  curvilinear. 

409.  These  considerations  are  easily  extended  to  space  of  three 
dimensions.     An  ellipsoid 

-,  +  r,  +  ^  =  1,  where  a  >  b  >  c, 
a^      b-      c^ 

has  six  real  foci  in  its  principal  planes;  two,  Si,  Si',  in  the  xy-plane,  on 
the  axis  of  x,  at  a  distance  OSi  =  Va^  —  ¥  from  the  center  0;  two, 
S2,  S-i,  in  the  yz-plane,  on  the  axis  of  y,  at  the  distance  OS2  =  Vb^  —  (? 
from  the  center;  and  two,  Si,  S3',  in  the  2x-plane,  on  the  axis  of  x,  at 
the  distance  0^3  =  Va^  —  &  from  the  center.  It  should  be  noticed 
that,  since  6  >  c,  we  have  OSi  >  OSi]  i.  e.  Si,  Si'  lie  between  S3,  &'  on 
the  axis  of  x. 

The  same  holds  for  hyperboloids. 

Two  quadric  surfaces  are  said  to  be  confocal  when  their  principal 
sections  are  confocal  conies.  Now  this  will  be  the  case  for  two  quadric 
surfaces  whose  semi-axes  are  Oi,  bi,  Ci,  and  02,  62,  C2,  if  the  directions  of 
their  axes  coincide  and  if 

Oi'  —  61^  =  a-r  —  bi^,     br  —  cr  =  bi^  —  d^,     Oi^  —  Ci^  =  a^^  —  c-^. 


300  KINETICS  [410. 

Writing  these  conditions  in  the  form 

ai  —  ar  =  b^-  —  bi^  =  c^  —  c-c,     say  =  X, 
we  find  ai  =  a^  +  X,  bi  =  b{'  +  X,  ci  =  c^  +  X.     Hence  the  equation 

^2  i;2  yl 

+  t^  +  :t^=1.  (20) 


a'-  +  X      62_^  X      c2  +  X 

where  X  is  a  variable  parameter,  represents  a  system  of  oonfocal  quadric 
surfaces. 

410.  As  long  as  X  is  algebraically  greater  than  —  c-,  the  equation 
(20)  represents  ellipsoids.  For  X  =  —  c^  the  surface  collapses  mto  the 
interior  area  of  the  ellipse  in  the  x?/-plane  whose  vertices  are  the  foci 
Si,  &■>!  and  aSs,  &z  .  For  as  X  approaches  the  hmit  —  &,  the  three  semi- 
axes  of  (20)  approach  the  limits  V  a^  —  c^,  VV^  —  c^,  0,  respectively. 
This  limiting  ellipse  is  called  the  focal  ellipse.  Its  foci  are  the  points 
Si,  Si',  since  a?  -  c"  -  {V-  -  c^)  =  a?  -  b\ 

When  X  is  algebraically  <  —  c-,  but  >  —  a^,  the  equation  (20)  repre- 
sents hyperboloids;  for  values  of  X  <  —  a-  it  is  not  satisfied  by  any  real 
points.  As  long  as  —  6-  <  X  <  —  c-,  the  surfaces  are  hyperboloids  of 
one  sheet.  The  limiting  surface  X  =  —  c-  now  represents  the  exterior 
area  of  the  focal  ellipse  in  the  rv-plane.  The  limiting  In^perboloid  of 
one  sheet  for  X  =  —  6^  is  the  area  in  the  2x-plane  bounded  by  the  hyper- 
bola whose  vertices  are  Si,  Si',  and  whose  foci  are  Ss,  S3'.  This  is  called 
the  focal  hyperbola. 

Finally,  when  —  a^  <  X  <  —  b-,  the  surfaces  are  hj'perboloids  of  two 
sheets,  the  limiting  hyperboloid  X  =  —  a^  collapsing  into  the  ?/2-plane. 

411.  It  appears  from  these  geometrical  considerations,  that  there 
are  passing  through  every  point  of  space  three  surfaces  confocal  to  the 
fundamental  ellipsoid  i^/a^  +  y'^/b^  +  z^/c"^  =  1  and  to  each  other,  viz.: 
an  ellipsoid,  a  hyperboloid  of  one  sheet,  and  a  hyperboloid  of  two  sheets. 
This  can  also  be  shown  analytically,  as  there  is  no  difficulty  in  proving 
that  the  equation  (20)  has  three  real  roots,  say  Xi,  X2,  X3,  for  every  set 
of  real  values  of  x,  y,  z,  and  that  these  roots  are  confined  between  such 
limits  as  to  give  the  three  surfaces  just  mentioned. 

The  quantities  Xi,  X2,  X3  can  therefore  be  taken  as  co-ordinates  of  the 
point  {x,  y,  z);  and  these  elliptic  co-ordinates  of  the  point  are,  geomet- 
rically, the  parameters  of  the  three  quadric  surfaces  passing  through 
the  point  and  confocal  to  the  fundamental  ellipsoid ;  while,  analytically, 


413.]     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     301 

they  are  the  three  roots  of  the  cubic  (20).  To  express  x,  y,  z  in  terms 
of  the  elhptic  co-ordinates,  it  is  only  necessary  to  solve  for  x,  y,  z  the 
three  equations  obtained  by  substituting  in  (20)  successively  Xi,  X2,  Xs 
for  X. 

412.  The  geometrical  meaning  of  the  parameter  X  will  appear  by 
considering  two  parallel  tangent  planes  tto  and  tta  (on  the  same  side  of 
the  origin),  the  former  (tto)  tangent  to  the  fundamental  ellipsoid 
x^la?  +  2/^/6^  +  z-jc^  =  1,  the  latter  (tta)  tangent  to  any  confocal  surface 
X  or  x^lia?  +  X)  +  y'^Hh'^  +  X)  +  z^l{c^  +  X)  =  1.  The  perpendiculars 
go,  ^A,  let  fall  from  the  origin  0  on  these  tangent  planes  tto,  tta,  are  given 
by  the  relations  (the  proof  being  the  same  as  in  Art.  399) 

go=  =  o?a^  +  6^/3^  +  c'y^  (21) 

qK2  =  (^2  +  x)a2  +  (62  +  X)/32  +  (c^  +  \)y\  (22) 

where  a,  /3,  7  are  the  direction  cosines  of  the  common  normal  of  the 
planes  tto,  tta.     Subtracting  (21)  from  (22),  we  find,  since  a-  +  /S^  +  y^ 

=  1, 

^A-  —  go-  =  X;  (23) 

i.  €.  the  parameter  X  of  any  one  of  the  confocal  surfaces  (20)  is  equal  to 
the  difference  of  the  squares  of  the  perpendiculars  let  fall  from  the  common 
center  on  any  tangent  plane  to  the  surface  X,  and  on  the  parallel  tayigenl 
plane  to  the  ftmdamental  ellipsoid  X  =  0. 

413.  Let  us  now  apply  these  results  to  the  question  of  the  distribu- 
tion of  the  principal  axes  throughout  space. 

We  take  the  centroid  G  of  the  given  body  as  origin,  and  select  as 
fundamental  ellipsoid  of  our  confocal  system  the  polar  reciprocal  of  the 
central  ellipsoid,  i.  e.  the  ellipsoid  (17)  formed  for  the  centroid,  for 
which  the  name  "fundamental  ellipsoid  of  the  body"  was  introduced  in 
Art.  401.     Its  equation  is 

qi'  ^  92^  ^93^  ' 

if  ?i,  ?2,  qs  are  the  principal  radii  of  inertia  of  the  body. 

The  radius  of  inertia  go  for  any  centroidal  lino  In  can  be  constructed 
(Art.  400)  by  laying  a  tangent  plane  to  this  ellipsoid  perpendicular  to 
the  line  k;  if  this  line  meets  the  tangent  plane  at  Qo  (Fig.  85),  then 


302  KINETICS  [414. 

Qo  =  GQo.     Analytically,  if  a,  /3,  7  be  the  direction  cosines  of  lo,  go  is 
given  by  formula  (21)  or  (12')- 

To  find  the  radius  of  inertia  q  for  a  line  I,  parallel  to  lo,  and  passing 
through  any  point  P,  we  lay  through  P  a  plane  tta,  perpendicular  to  I, 
and  a  parallel  plane  wo,  tangent  to  the  fundamental  ellipsoid;  let  Qk, 


Fig.  85. 


Qo  be  the  intersections  of  these  planes  with  the  centroidal  line  k.    Then, 
putting  GQo  =  qo,  GQk  =  q\,  GP  =  r,  PQ\  =  d,  we  have,  by  Art.  385, 

q^  =  qa'  +  d?. 

The  figure  gives  the  relation  d^  =  r^  —  q\^,  which,  in  combination  with 

(23)  reduces  the  expression  for  the  radius  of  inertia  for  the  line  I  to 

the  simple  form 

g2  =  r2  -  X.  (24) 

414.  The  value  of  r^  —  X,  and  hence  the  value  of  q,  remains  the  same 
for  the  perpendiculars  to  all  planes  through  P,  tangent  to  the  same 
quadric  surface  X:  these  perpendiculars  form,  therefore,  an  equimo- 
mental  cone  at  P.  By  varying  X  we  thus  obtain  all  the  equimomental 
cones  at  P.  The  principal  diameters  of  all  these  cones  coincide  in 
direction,  since  they  coincide  with  the  directions  of  the  principal  axes 
of  the  momental  ellipsoid  at  P  (see  Art.  396) ;  but  they  also  coincide  with 
the  principal  diameters  of  the  cones  enveloped  by  the  tangent  planes  tta. 
It  thus  appears  that  the  principal  axes  at  the  point  P  coincide  in  direction 


414.]     MOMENTS  OF  INERTIA  AND  PRINCIPAL  AXES     303 

with  the  principal  diameters  of  the  tangent  cone  from  P  as  vertex  to  the 
fundamental  ellipsoid  x^jq^  +  y'^lq-^  +  zV?3^  =  1- 

Instead  of  the  fundamental  ellipsoid,  we  might  have  used  any 
quadric  surface  X  confocal  to  it.  In  particular,  we  may  select  the  con- 
focal  surfaces  Xi,  X2,  X3  that  pa  s  through  P.  For  each  of  these  the  cone 
of  the  tangent  planes  collapses  into  a  plane,  viz.  the  tangent  plane  to 
the  surface  at  P,  while  the  cone  of  the  perpendiculars  reduces  to  a  single 
line,  viz.  the  normal  to  the  surface  at  P.  Thus  we  find  that  the  prin- 
cipal axes  at  amj  point  P  coincide  in  direction  with  the  normals  to  the 
three  quadric  sinfaces,  confocal  to  the  fundamental  ellipsoid  and  passing 
through  P. 

For  the  magnitudes  of  the  principal  radii  qx,  qy,  qz,  at  P,  we  evidently 
have 

qi    =   7-2   _  y^^^       q2    =   j-l   —  X2,       5^2    =   j-2    —   Xj. 


CHAPTER  XVII. 
RIGID  BODY  WITH  A   FIXED  AXIS. 

415.  A  rigid  body  with  a  fixed  axis  has  but  one  degree  of 
freedom.  Its  motion  is  fully  determined  by  the  motion  of 
any  one  of  its  points  (not  situated  on  the  axis),  and  any  such 
point  must  move  in  a  circle  about  the  axis.  Any  particular 
position  of  the  body  is,  therefore,  determined  by  a  single 
variable,  or  co-ordinate,  such  as  the  angle  of  rotation.  Just 
as  the  equilibrium  of  such  a  body  depends  on  a  single  con- 
dition (see  Art.  234),  so  its  motion  is  given  by  a  single 
equation. 

This  equation  is  obtained  at  once  by  "  taking  moments 
about  the  fixed  axis."  For,  according  to  the  proposition  of 
angular  momentum  (Art.  360),  the  time-rate  of  change  of 
angular  momentum  about  any  axis  is  equal  to  the  moment 
of  the  external  forces  about  this  axis.  Hence,  denoting  this 
moment  by  H  and  taking  the  fixed  axis  as  axis  of  z,  we  have 
as  equation  of  motion  the  last  of  the  equations  (3'),  Art.  360, 
viz., 

—  2w(x?/  -  yx)  =  H.  (1) 

416.  The  angular  momentum,  1,m{xy  —  yx),  about  the  fixed 
axis  can  be  reduced  to  a  more  simple  form.  For  rotation  of 
angular  velocity  co  about  the  2-axis  we  have  (Art.  48,  Ex.  1) 
X  =  —  coy,  y  =  o)X,  so  that 

'2m{xy  —  yx)  =  (jo1,ni{x'^  +  y'')  =  w'Smr^  =  /co. 
304 


417.]  RIGID  BODY  WITH  A  FIXED  AXIS  805 

where  r  is  the  distance  of  the  particle  m  from  the  axis  and 
/  =  Swr^  the  moment  of  inertia  of  the  body  for  this  axis. 

This  expression  for  the  angular  momentum  can  be  derived 
without  reference  to  any  co-ordinate  system.  For  evidently 
mcor  is  the  linear  momentum  of  the  particle  m,  mcor^  is  its 
moment,  i.  e.  the  angular  momentum  of  the  particle,  about 
the  axis;  andZwiwr-  =  uZmr-  =  /co  is  the  angular  momentum 
of  the  body  about  the  axis. 

It  thus  appears  that,  just  as  in  translation  the  linear  mo- 
mentum of  a  body  is  the  product  of  its  mass  into  its  linear 
velocity,  so  in  the  case  of  rotation  the  angular  momentum 
of  the  body  about  the  axis  of  rotation  is  the  product  of  its  moment 
of  inertia  (for  this  axis)  into  the  angular  velocity. 

As  regards  the  right-hand  member  of  equation  (1),  the 
reactions  of  the  axis  need  not  be  taken  into  account  in  forming 
the  moment  H;  for  as  these  reactions  meet  the  axis,  their 
moments  about  this  axis  are  zero. 

417.  Substituting  7co  for  Xm(xy  —  yx)  in  equation  (1),  and 
observing  that  the  moment  of  inertia  I  about  a  fixed  axis 
remains  constant,  we  find  the  equation  of  motion  in  the  form 

/1  =  H;  (2) 

^.  e.  for  rotation  about  a  fixed  axis  the  product  of  the  moment  of 
inertia  for  this  axis  into  the  angular  acceleration  equals  the 
moment  of  the  external  forces  about  this  axis;  just  as,  in  the  case 
of  rectilinear  translation,  the  product  of  the  mass  of  the  body 
into  the  linear  acceleration  equals  the  resultant  force  R  along 

the  line  of  motion: 

dv 
m-TT  =  K. 
dt 

And  just  as  the  latter  equation  may  serve  to  determine 
21 


306  KINETICS  [418. 

experimentally  the  mass  of  a  body  by  observing  the  accelera- 
tion produced  in  it  by  a  given  force  R,  e.  g.  the  force  of 
gravity  (as  in  the  gravitation  system,  Art.  177),  so  the  former 
equation,  (2),  may  serve  to  determine  experimentally  the 
moment  of  inertia  of  a  body  about  a  line  I,  by  observing  the 
angular  acceleration  produced  in  the  body  when  rotating 
about  I  under  given  forces. 

418.  For  the  kinetic  energy  of  a  body  rotating  with  angular 
velocity  co  about  any  axis  we  have 

T  =  '^hnv^  =  Hhnoi^r^  =  i/w^,  (3) 

an  expression  which  is  again  similar  in  form  to  that  for  the 
kinetic  energy  of  a  body  in  translation,  viz.  T  =  ^mv-. 

When  the  axis  is  fixed  so  that  I  is  constant,  the  equation 
of  motion  (2) ,  multiplied  by  co  and  integrated,  say  from  t  =  0 
to  t  =  t,  gives  the  relation 

i/co2  -  i/coo^  =  £'Ho}dt,  (4) 

which  expresses  the  principle  of  khietic  energy  and  work. 

419.  As  an  example  consider  the  compound  pendulum,  i.  e. 
a  rigid  body  with  a  fixed  horizontal  axis  and  subject  to  gravity 
alone.  If  OG  =  h  is  the  distance  of  the  centroid  G  from  the 
fixed  axis  and  6  the  angle  made  by  OG  with  the  vertical 
plane  through  the  axis  we  have  H  =  Mgh  sin5.  Denoting 
by  q  the  radius  of  inertia  about  the  centroidal  axis  through  G 
parallel  to  the  fixed  axis  so  that  the  moment  of  inertia  about 
the  fixed  axis  is  =  M{q-  +  h^),  we  find  the  equation  of 
motion  (2)  in  the  form 

q'  +  h 


sine.  (5) 


Comparing  this  with  the  equation  of  motion  of  the  simple 


420.]  RIGID  BODY  WITH  A  FIXED  AXIS  307 

pendulum  (Arts.  63,  335),  6  =  —  (g/l)  sin^,  it  appears  that 
the  motion  of  a  compound  pendulum  is  the  same  as  that  of  a 
simple  pendulum  of  length 

l-h  +  ^j.  (6) 

This  is  called  the  length  of  the  equivalent  simple  pendulum. 
The  foot  0  of  the  perpendicular  let  fall  from  the  centroid 
G  on  the  fixed  axis  is  called  the  center  of  suspension.  If  on  the 
line  OG  a  length  OC  =  I  be  laid  off,  the  point  C  is  called  the 
center  of  oscillation.  It  appears,  from  (6),  that  G  lies  between 
0  and  C. 

The  relation  (6)  can  be  written  in  the  form 

h(l  -  h)  =  cf,  or  OG-GC  =  const. 

As  this  relation  is  not  altered  by  interchanging  0  and  C,  it 
follows  that  the  centers  of  oscillation  and  suspension  are  inter- 
changeable; i.  e.  the  period  of  a  compound  pendulum  remains 
the  same  if  it  be  made  to  swing  about  a  parallel  axis  through 
the  center  of  oscillation. 

420.  Exercises. 

(1)  A  pendulum,  formed  of  a  cylindrical  rod  of  radius  a  and  length 
L,  swings  about  a  diameter  of  one  of  the  bases.  Find  the  time  of  a 
small  oscillation. 

(2)  A  cube,  whose  edge  is  a,  swings  as  a  pendulum  about  an  edge. 
Find  the  length  of  the  equivalent  simple  pendulum. 

(3)  A  circular  disk  of  radius  r  revolves  uniformly  about  its  axis, 
making  100  rev./min.     What  is  its  kinetic  energy? 

(4)  A  homogeneous  straight  rod  of  length  I  is  hinged  at  one  end  so 
as  to  turn  freely  in  a  vertical  plane.  If  it  be  dropped  from  a  horizontal 
position,  with  what  angular  velocity  does  it  pass  through  the  ^  vertical 
position?     (Equate  the  kinetic  energy  to  the  work  of  gravity.) 

(5)  A  homogeneous  plate  whoso  shape  is  that  of  the  segment  of  a 
parabola  bounded  by  the  curve  and  its  latus  rectum  swings  about  the 


308 


KINETICS 


1421, 


latus  rectum  which  is  horizontal.     Find  the  length  of  the  equivalent 
simple  pendulum. 

(6)  When  q  is  given  while  I  and  h  vary,  the  equation  (6)  represents 
a  hyperbola  whose  asymptotes  are  the  axis  of  I  and  the  bisector  of  the 
angle  between  the  (positive)  axes  of  h  and  I.  Show  that  Imia  =  2q  for 
h  =  q;  also  that  I,  and  hence  the  period  of  oscillation,  can  be  made  very 
large  by  taking  h  either  very  large  or  very  small.  The  latter  case  occurs 
for  a  ship  whose  mclacenter  (which  plays  the  part  of  the  point  of  suspen- 
sion) Ues  very  near  its  centroid. 

(7)  A  homogeneous  circular  disk,  1  ft.  in  diameter  and  weighing 
25  lbs.,  is  making  240  rev./min.  when  left  to  itseK.  Determine  the 
constant  tangential  force  applied  to  its  rim  that  would  bring  it  to  rest 
in  1  min. 

421.  While  a  single  equation  determines  the  motion  of  a 
body  with  a  fixed  axis,  the  other  five  equations  of  motion  of  a 
rigid  body  must  be  used  to  determine  the  reactions. 

The  axis  will  be  fixed  if  any  two  of 
its  points  A,  5  are  fixed.  The  reac- 
tion of  the  fixed  point  A  can  be  re- 
solved into  three  components  Ax,  Ay, 
Az,  that  of  B  into  B^,  By,  B^.  By 
introducing  these  reactions  the  body 
becomes  free;  and  the  system  com- 
posed of  these  reactions,  of  the  exter- 
nal forces,  and  of  the  reversed  effec- 
tive forces  must  be  in  equilibrium. 
We  take  the  axis  of  rotation  as  axis 
of  z  (Fig.  86)  so  that  the  z-co-ordinates 
of  the  particles  are  constant,  and  hence  i  =  0,  S  =  0;  and 
we  put  OA  =  a,  OB  =  h.  Then  the  six  equations  of  mo- 
tion are  (see  Art.  359  (2)  and  Art.  360  (3)): 

^mx  =  SX+  ^.  +  B„ 

Zmij      =      37       -\-      Ay+      By, 


Fig.  86. 


423.]  RIGID  BODY  WITH  A  FIXED  AXIS  309 

0  =  SZ    -\-  A,  -\-  B,, 
—  'Zmzij  =  i:(yZ  -  zY)  -  aAy  -  hBy, 
llmzx  =  l^izX  —  xZ)  +  a^^  +  bB^, 
'Zm{xy  —  yx)  =  2(xF  —  yX). 

422.  It  remains  to  introduce  into  these  equations  the  values 
for  X,  y.  As  the  motion  is  a  pure  rotation,  we  have  (see  Art. 
48,  Ex.  1)  X  =  —  ooy,  y  =  oox;  hence,  x  =  —  6:y  —  oi^x, 
y  =  6)x  —  co^y.     Summing  over  the  whole  body,  we  find 

2mf  =  —  (jiZmy  —  w-^mx  =  —  Mioy  —  Mw'^x, 
Xmij  =       (li^mx  —  w'^^my  =       Miox  —  Mco^y, 

where  x,  y  are  the  co-ordinates  of  the  centroid;  and 

-  ^mzij  =  —  ooliffizx  +  co^'^niyz  =  —  E6:  +  -Dco^, 
l^mzx  =  —  ulimyz  —  co^lmzx  =  —  Du  —  Eo:"^, 

Xm{xij  —  yx)  =  coSrwo:"  —  (xr^mxy  +  os'^my^  +  co-Zmxy  =  Co), 
where  C  =  1,m(x-  -{-  y-),  D  =  Zmyz,  E  =  l^mzx  are  the  no- 
tations introduced  in  Art.  387. 

With  these  values  the  equations  of  motion  assume  the  form : 

-  MxiJ"  -  MyCi  =  SX  +  A,  -1-  B^, 

-  Myo)"-  +  Mx(h  =  1:Y  +  Ay+  By, 

0  =  2Z   +  ^.  +  B,, 
Dco2  -  E(h  =  i:OjZ  -  zY)  -  aAy  -  hBy,        ^  ' 
-  Eoi"-  -  Z)w  -  ^{zX  -  xZ)  +  aA,  +  hB„ 
C(h  =  Z(xY  -  yX). 

423.  The  last  equation  is  identical  with  equation  (2),  Art. 
417. 

The  components  of  the  reactions  along  the  axis  of  rotation 
occur  only  in  the  third  equation  and  can  therefore  not  be 
found  separately.     The  longitudinal  pressure  on  the  axis  is 

=  -  A.-  B,  =  2Z. 


310  •  KINETICS 


[424. 


The  remaining  four  equations  are  sufficient  to  determine 

■^x>    -^Vl    ^x,    ijy 

The  total  stress  to  which  the  axis  is  subject,  instead  of 
being  represented  l)y  the  two  forces,  at  A  and  B,  can  be 
reduced  for  the  origin  0  to  a  force  and  a  couple.  The  equa- 
tions (7)  give  for  the  components  of  the  force 

-  A,-  B,  -  SX  +  Mxco2  +  My6i, 

-  Ay  -  By  ^  ^Y  +  Myoi''  -  Ma-w,  (8) 

-  A,  -  B,  =  SZ. 

This  force  consists  of  the  resultant  of  the  external  forces, 


E  =  V(SX)2  +  (SF)2  +  (SZ)^ 

and  two  forces  in  the  a;?/-plane  which  form  the  reversed  effec- 
tive force  of  the  centroid;  for  Mxcji^  and  Myw-  give  as  re- 
sultant the  centrifugal  force  Mco^V^^  +  y~  =  Mw-f,  directed 
from  the  origin  towards  the  projection  of  the  centroid  on  the 
a;?/-plane,  while  Myic,  —  Mx<l}  form  the  tangential  resultant 
ilfcof,  perpendicular  to  the  plane  through  axis  and  centroid. 
The  couple  has  a  component  in  the  ^2-plane,  and  one  in  the 
zx-plane,  viz.: 

aAy  +  bBy  =  ^{yZ  -  2F)  -  Dco^  +  E'ci, 
-  a  A,  -  hB,  =  Z{zX  -  zZ)  +  iJw^  +  Deb,  ^^ 

while  the  component  in  the  rr?/-plane  is  zero.  The  resultant 
couple  lies,  therefore,  in  a  plane  passing  through  the  axis  of 
rotation. 

424.  In  the  particular  case  lohen  no  forces  X,  Y,  Z  are 
acting  on  the  body,  the  last  of  the  equations  (7),  or  equation 
(2),  shows  that  the  angular  velocity  co  remains  constant.  The 
stress  on  the  axis  of  rotation  will,  however,  exist;  and  the 
axis  will  in  general  tend  to  change  both  its  direction,  owing 
to  the  couple  (9),  and  its  position,  owing  to  the  force  (8). 


425.]  RIGID  BODY  WITH  A  FIXED  AXIS  311 

If  the  axis  be  not  fixed  as  a  whole,  but  only  one  of  its 
points,  the  origin,  be  fixed,  the  force  (8)  is  taken  up  by  the 
fixed  point,  while  the  couple  (9)  will  change  the  direction 
of  the  axis.  Now  this  couple  vanishes  if,  in  addition  to 
the  absence  of  external  forces,  the  conditions 

D  =  Zmyz  =  0,   E  =  Zmzx  =  0  (10) 

are  fulfilled.  In  this  case  the  body  would  continue  to  rotate 
about  the  axis  of  z  even  if  this  axis  were  not  fixed,  provided 
that  the  origin  is  a  fixed  point.  A  line  having  this  property 
is  called  a  permanent  axis  of  rotation. 

As  the  meaning  of  the  conditions  (10)  is  that  the  axis  of 
z  is  a  principal  axis  of  inertia  at  the  origin  (see  Art.  395),  we 
have  the  proposition  that  if  a  rigid  body  with  a  fixed  point, 
not  acted  wpon  by  any  forces,  begin  to  rotate  about  one  of  the 
principal  axes  at  this  jjoint,  it  will  continue  to  rotate  uni- 
formly about  the  same  axis.  In  other  words  the  principal 
axes  at  any  point  are  always,  and  are  the  only,  permanent 
axes  of  rotation.  This  can  be  regarded  as  the  dynamical 
definition  of  principal  axes. 

425.  It  appears  from  the  equations  (8)  that  the  position  of 
the  axis  of  rotation  will  remain  the  same  if,  in  addition  to  the 
absence  of  external  forces,  the  conditions 

x  =  0,     7j  =  0  (11) 

be  fulfilled ;  for  in  this  case  the  components  of  the  force  (8)  all 
vanish.  If,  moreover,  the  axis  of  rotation  be  a  principal 
axis,  the  rotation  will  continue  to  take  place  about  the  same 
line  even  when  the  body  has  no  fixed  point. 

The  conditions  (11)  moan  that  the  centroid  lies  on  the 
axis  of  2;  and  it  is  known  (Art.  395)  that  a  centroidal  principal 
axis  is  a  principal  axis  at  every  one  of  its  points.     The  axis 


312  KINETICS 


[425. 


of  z  must  therefore  be  a  principal  axis  of  the  body,  i.  e.  a 
principal  axis  at  the  centroid.  We  have  thus  the  proposi- 
tion: //  a  free  rigid  body,  not  acted  upon  by  any  forces,  begin 
to  rotate  about  one  of  its  centroidal  principal  axes,  it  will  con- 
tinue to  rotate  uniformly  about  the  same  line. 


CHAPTER  XVIII. 
RIGID  BODY  WITH  A  FIXED  POINT. 

1.  The    general    equations    of   motion. 

426.  If  the  fixed  point  0  be  taken  as  origin  and  the  reac- 
tion at  0  be  denoted  by  A  (as  in  Art.  233)  the  equations  of 
motion  (2),  (3)  of  Arts.  359,  360  become: 

2mi:  =  SX+^x,     2m?/  =  SF+Aj,,     Sw2=  2Z  +  A„    (1) 

Sm(y3  -  zy)  =  ZiyZ  -  zY),      i:m{zx  -  xz)  =  2(2X  -  xZ), 

Xmixy  -  yx)  =  i:{xY  -  yX).  (2) 

The  equations  (1)  merely  serve  to  determine  the  reaction 
A,  while  the  equations  (2)  determine  the  motion.  There 
should  be  three  such  equations  because  a  rigid  body  with  a 
fixed  point  has  three  degrees  of  freedom  (Art.  233) 

Kinematically,  the  instantaneous  state  of  motion  is  a 
rotation  about  an  axis  through  0  and  is  given  by  the  rotor 
CO  (Arts.  116,  128).  The  course  of  the  motion  consists  of 
the  rolling  of  the  cone  of  body  axes  over  the  cone  of  space 
axes  (Art.  131). 

Dynamically,  the  instantaneous  state  of  motion  of  the 
body  is  given  by  the  impulse-vector  h  (Art.  367)  which  is  the 
resultant  of  the  angular  momenta  of  all  the  particles  con- 
stituting the  body,  or  (Arts.  372,  373)  the  vector  of  that 
impulsive  couple  which,  acting  on  the  body  at  rest,  would 
impart  to  it  its  instantaneous  state  of  motion,  i.  e.  would 
produce  instantaneously  the  rotor  cj.  The  given  external 
forces  reduce  to  a  resultant  R  through  0,  which  is  taken 

313 


314  KINETICS 


[427. 


up  by  the  fixed  point  and  does  not  affect  the  motion,  and 
a  couple,  of  vector  H,  whose  components  are  the  right- 
hand  members  of  (2).  Writing  these  eciuations  in  the  form 
(3")  of  Art.  361,  viz. 

dt        ^^'      dt        ^"      dt        ^^'  ^^^ 

we  see  that  the  time-rate  of  change  of  the  vector  h  is  geometric- 
ally equal  to  the  vector  H. 

The  main  ciuestion  is  the  relation  between  the  vectors  co 
and  h. 

427.  Now  for  the  angular  momentum  about  the  axis 
Ox  we  have  since  x  =  ooyZ  —  w.y,  y  =  WzX  —  w^z,  z  =  co^y 
-  coyx  (Art.  118): 

hx  =  2w(?/i  —  zy)  =  a)xS??i(?/-  +  z^)  —  Oylmxy  —  w.'^mzx, 

or,  with  the  notation  of  Art.  387,  hjc  =  Ao)^  —  Fcoy  —  Eo^z. 
Determining  hy,  h-  in  the  same  w^ay  we  find: 

hx  =       Acox  —  Fcoy  —  Ewz, 

hy  =  —  Fu)x  -\-  Boiy  —  Dcaz,  (3) 

hz  =   —  Ewx  —  DcOy  +  CcOz. 

These  equations  enable  us  to  find  the  vector  h  Avhcn  co  is 
given,  and  vice  versa.  The  relation  between  these  vectors 
which  are  evidently  in  general  not  parallel  appears  from  the 
equation  of  the  momental  ellipsoid,  (10),  Art.  392.  If  we 
select  the  arbitrary  constant  e  so  that  this  ellipsoid  passes 
through  the  extremity  of  the  rotor  co,  that  is  so  that 

AcOx"  +  Bcoy^  +  Cwr  -  27)ajyCO,  -  2Ec^zC^x  -  2FwxOiy  =  iWe^ 

it  appears  that  hx,  hy,  hz  are  one  half  the  partial  deriva- 
tives of  the  left-hand  member  of  this  equation,  and  hence 


429.]  RIGID  BODY  WITH  A  FIXED  POINT  315 

the  vector  h  is  normal  to  the  tangent  plane  to  the  momental 
ellipsoid  at  the  extremity  of  w;  in  other  words,  the  plane  of 
the  impulsive  couple  h  is  conjugate  to  the  direction  of  co  with 
respect  to  the  momental  ellipsoid. 

428.  For  the  kinetic  energy  we  have  if  r  is  the  distance 
of  the  particle  m  from  the  instantaneous  axis  co: 

T  =  i:hnv~  =  i/w",  (4) 

where  /  =  w//ir-  is  the  moment  of  inertia  of  the  body  aljout 
the  instantaneous  axis.  Now  if  a,  ^,  y  are  the  direction 
cosines  of  this  axis,  i.  e.  of  the  rotor  oo,  we  have  by  (9), 
Art.  390, 

I  =  Aa"-  -\-  /^^2  +  c^".  _  oD^y  _  2Eya  -  2/'«/3; 
multiplying  by  tco^  we  find 

It  follows  by  (3)  that 

7        dT  dT  dT  ,., 

doox  aoiy  oiOz 

Multiplying  (3)  by  w^:,  oiy,  co^  and  adding  we  find 

hxo^x  +  hyWy  +  hzWz  —  2T,  (7) 

which  means  that  the  kinetic  energy  is  one  half  the  dot-product 
h  •  w  of  the  vectors  h  and  co. 

429.  All  these  relations  become  far  more  simple  if  we  take 
as  axes  of  co-ordinates  the  principal  axes  at  0;  but  it  must 
be  kept  in  mind  that  these  are  rnoving  axes.  Distinguishing, 
as  in  Kinematics,  components  along  moving  axes  by  the 
subscripts  1,  2,  3  instead  of  x,  y,  z,  and  (Unioting  the  principal 
moments  of  inertia  at  0  by  /i,  h,  h  (Art.  393)  we  have 
by  (3) 


316  KINETICS  [430. 

hi  =  /icoi,     /i2  =  I2OJ2,    hi  =  /3CO3.  (8) 

These  equations  show  that  if  the  vector  of  an  impulsive 
couple  is  parallel  to  a  principal  axis  at  0,  it  produces  an 
angular  velocity  about  this  axis;  it  follows  from  the  equations 
(3)  that  the  condition  is  not  only  sufficient  but  necessary. 
Comp.  Art.  424. 

For  the  kinetic  energy  we  have  by  (12),  Art.  393: 

=  i(/iCOi^  +  /2CO22  +  Jscoa^) 

Substituting  for  7i,  h,  h,  or  for  coi,  C02,  cos  their  values  from 
(8)  we  find 

=  hicoi  +  hoc^o  +  hzws  (9) 

^hl     }i2^      Jil_ 
Ii       h'^  h   ' 

430.  Euler's  Equations.  It  appears  from  the  equations 
(2')  that  the  impulse  h  which,  by  (3)  or  (8),  determines  co  and 
hence  the  instantaneous  state  of  motion  of  the  body,  varies 
in  the  course  of  the  motion,  under  the  action  of  the  external 
forces  both  in  magnitude  and  in  direction,  and  also  both  rela- 
tively to  the  body  and  relatively  to  the  fixed  trihedral  of  axes. 

It  is  generally  found  most  convenient  to  determine  first 
the  variation  of  the  vector  h  relatively  to  the  moving  axes, 
and  then  to  determine  the  motion  of  the  trihedral  of  the 
moving  axes  with  respect  to  the  fixed  axes.  The  former 
of  these  problems  is  solved  by  Euler's  equations  (Art.  432) 
while  the  latter  can  be  solved  with  the  aid  of  Euler's  angles 
(Art.  434)  or  any  other  suitable  parameters. 

431.  Euler's  equations  are  essentially  the  equations  (2') 


433.1  RIGID  BODY  WITH  A  FIXED  POINT  317 

when  referred  to  the  principal  axes  at  0;  they  express  the 
geometrical  relation  dhjdt  —  H. 

The  variation  of  the  vector  h  (drawn  from  0)  depends  on 
the  motion  of  its  extremity  whose  co-ordinates  are  h^,  hy,  hz 
with  respect  to  the  fixed  axes,  and  hi,  hi,  hs  with  respect  to 
the  moving  axes  (for  the  present,  not  necessarily  the  principal 
axes).  The  absolute  velocity  (/t^,  hy,  h^)  of  the  extremity  of 
h  can  be  resolved  into  its  relative  velocity  ih],  h-i,  A3)  and  the 
body-velocity  co  X  h  (Arts.  118,  119,  129)  whose  components 
along  the  moving  axes  are  wo/is  —  cojin,  coshi  —  cojis,  coi/^o  — 
ojo/ii.  The  equations  (2')  referred  to  any  moving  axes  fixed 
in  the  body  become  therefore 

-^r  +  oo^h  —  usho  =  Hi, 


—  -f  Ojjli  —  (jOih  =  H2,  (10) 


-17  -f  C01/12  —  C02/11  =  ^3; 

or  briefly,  in  vector  form :  h  -\-  uXh  =  H 

432.  If,  in  particular,  we  take  as  moving  axes  the  principal 
axes  at  0,  the  equations  (10),  owing  to  the  relations  (8), 
reduce  to  the  following: 

Il<j^l   +    (^3    —    /2)W2C03    =    H],        /oWo   +    (/l    —    73)C03C01    =    Ho, 

/3CO3   +    (^2    —    /l)wia!2    =    H3,  (11) 

which  are  known  as  Euler's  equations  of  motion  of  a  rigid 
body  with  a  fixed  point.  Their  integration  gives  coi,  C02,  ws, 
and  hence  w,  as  functions  of  the  time  t. 

433.  Analytically,  the  equations  (10)  can  be  derived  from  the  equa- 
tions (2),  Art.  426,  or  rather  from  the  corresponding  equations  for  fixed 
axes  coinciding  at  the  instant  considered  with  the  moving  axes,  viz. : 


318 


KINETICS 


[434. 


S7re(2/i2i  -  2i^i)  =  Hi,     S?n(2ix-i  -  XiZi)  =  H2,     Sm  {xiyi  —  yai)  =  Hz, 
by  introducing  for  Xi,  iji,  zi  their  values  from  (4'),  Art.  141. 
We  thus  find  for  Xm{yiZi  —  Ziyi): 

m(o:i'^7nxiyi  +  oiiEmy^  +  us'StnyiZi)  —  d^'^mijiZi 

—  co2(cow?H2iXi  +  ui^myiZi  +  wsStozi^)  +  w^ZmyiZi 

+  ojiS?h(2/i^  +  2i^)  —  ui'LviXiyi  —  wsS/ziZiXi, 

or  with  the  notation  of  Art.  387: 

oii{Fui  +  Ca)2  +  Dois)   —  oJiiEo^i  +  Duo  +  ZJws)   +  Aui  —  F6j2  —  Eio3 

—  W2( — Ed}]    —  Z)c02  +  Ccos)    —  C03( —  FcOl    -{-  BiOo    —   Dui)    -{-  Awi — Foil — Ecili 


=  wohs  —  coaho  -\- 


dhi 

Hi 


by  (3),  Art.  427.  The  relations  (3)  hold  of  course  for  mo\'ing  axes  as 
well  as  for  fixed  axes.  But  for  the  fixed  axes  the  coefficients  of  u>x,  coy,  uz, 
i.  e.  the  moments  and  products  of  inertia  for  the  fixed  axes,  are  not 
constant,  while  for  the  mo\'ing  axes  the  coefficients  of  wi,  002,  W3  are 
constant. 

434.  The  position  of  the  moving  trihedral  at  any  instant 
with  respect  to  the  fixed  trihedral  can  be  assigned  bj^  three 

angles  as  follows.  Let 
X,  Y,  Z  (Fig.  87)  be  the 
intersections  of  the  fixed 
axes,  Xi,  Fi,  Zi  those  of 
the  moving  axes  Avith 
the  sphere  of  radius  I  de- 
scribed about  the  fixed 
point  0;  and  let  N  be. 
the  intersection  with  the 
same  sphere  of  the  nodal 
line,  or  line  of  nodes,  i.  e. 
the    line    in   which    the 


1 

t 

^x)^' 

\ 

\e 

0' 

\""""A 

A 

^-^  \  y     xY 

Fig.  87. 


planes  XOY  and    XiOFi  intersect.     Then  the  angles  ZZi 
=  6,  NXi  =  <p,  XN  =  4^,  usually  called  Euler's  angles,  fully 


435.]  RIGID  BODY  WITH  A  FIXED  POINT  319 

determine  the  relative  position  of  one  trihedral  with  respect 
to  the  other.  If  the  moving  trihedral  be  initially  coincident 
with  the  fixed  trihedral  it  can  be  carried  into  any  other  position 
in  three  steps:  (a)  turn  the  trihedral  XiYiZi,  when  coinci- 
dent with  XYZ,  about  OZ  counterclockwise  until  OXi  coin- 
cides with  the  assumed  positive  sense  of  the  nodal  line  ON, 
and  call  the  angle  of  this  rotation  xp;  (6)  in  the  new  posi- 
tion turn  XiFiZi  counterclockwise  about  ON  until  the  plane 
XiOYi  falls  into  its  final  position,  the  angle  of  this  rotation 
is  6;  (c)  finally  turn  XiYiZi  about  OZi  counterclockwise 
through  an  angle  ^  until  OXi  reaches  its  final  position. 

435.  The  rotor  co  can  evidently  be  resolved  along  the  axes 
ON,  OZi,  OZ  into  the  components  6,  <p,  xj/;  hence  the  sum 
of  the  projections  of  these  components  6,  <jp,  xp  on  OXi  must 
be  equal  to  coi;  similarly  for  co2,  C03.  As  Fig.  87  shows,  the 
direction  cosines  of  ON,  OZy,  OZ  with  respect  to  the  moving 
trihedral  are 

Z, 

0 


Hence 


Xr 

Y, 

N 

CO'&ip 

—  sm.^p 

Z: 

0 

0 

1 

Z  sin0  sin^     sin0  cos^     cos0 

ui  =  d  cos^  +  xp  sin0  sin^, 

aj2  =  —  ^  sinv  +  ^  sin0  coS(p,  (12) 

W3  =    <^  +  '/'  COS0. 

By  substituting  these  values  in  Euler's  equations  we  obtain 
differential  equations  of  the  second  order  for  d,  ^p,  \p.  If 
Euler's  equations  have  been  solved  so  that  coi,  coo,  C03  have 
been  found  as  functions  of  t,  the  equations  (12)  arc  differential 
equations  of  the  first  order  for  6,  cp,  \p.  Solving  these  equa- 
tions for  d,  <p,  \p  we  have: 


320 


KINETICS 


[436. 


P   =   COi  COStp  —  CO2  SlUip, 

<p  =  —  ui  sintp  cotO  —  C02  cos(p  cot0  +  C03, 

)/'   =  COi  SlUip  CSC0  +  CO2  COS^  CSC0. 


(12') 


2.  Motion  without  forces. 

436.  Let  a  rigid  bod}-  with  a  fixed  point  0  be  given  an 
initial  angular  velocity  about  an  axis  through  0,  and  let  the 

resultant  couple  H  of   the 
^  /  external  forces  be  zero.  By- 

Art.  427,  the  initial  position 
of  the  bod}^  i.  e.  of  its  mo- 
mental  ellipsoid,  together 
with  the  initial  axis  of  rota- 
tion, determines  the  initial 
direction  of  the  impulse  h, 
this  direction  being  perpen- 
dicular to  the  tangent  plane 
to  the  ellipsoid  at  the  point 
P   where  it  is  met  by  the 


Fig.  88. 


instantaneous  axis  (Fig.  88). 

As  H  is  zero,  it  follows  from  (2'),  Art.  426,  that  h  is  constant 
in  magnitude  and  direction.  Moreover,  by  (9),  Art.  429, 
the  kinetic  energy  T  is  constant.  Finally,  it  can  be  shown 
that  the  perpendicular  5  let  fall  from  0  on  the  tangent  plane 
at  P  is  constant. 

To  prove  this  let 


Iixi^  4-  hyi^  +  hz,^ 


1 


be  the  equation  of  the  momental  ellipsoid  referred  to  the 
principal  axes  so  that  the  tangent  plane  at  P  (^,  77,  f)  has 
the  equation 

IiXi^  +  hyiV  +  hzit  =  1- 


438.1  RIGID  BODY  WITH  A  FIXED  POINT  321 

If  p  be  the  radius  vector  OP  of  P  we  have 
^  =  ^  =  ^  =^. 

OJl  Oi2  W3  OJ 

Hence 

by  (8),  Art.  429.     On  the  other  hand,  as  P  lies  on  the  elHpsoid 
we  have 

Ii^  +  W  +  h^^  =  1,    i'  e.    4  (^I'^i'  +  ^2052^  +  /3C032)  =  1. 

CO 

By  (9),  Art.  429,  this  shows  that  p/co  =  1/  Af2T.     Hence 

1  =  -i= 

5      V2r' 

and  as  both  h  and  T  are  constant,  5  is  constant. 

From  the  relation  between  the  directions  of  00  and  h  and 
the  constancy  of  h  and  5  it  follows  that  the  motion  of  the 
body  consists  in  the  rolling  of  its  momental  ellipsoid  over  a 
fixed  tangent  plane. 

437.  The  points  where  the  instantaneous  axis  meets  the 
momental  ellipsoid  form  a  curve,  fixed  in  the  body  and  moving 
with  it,  which  is  called  the  polhode  (path  of  the  pole  P). 
The  intersections  of  the  instantaneous  axis  with  the  fixed 
tangent  plane  form  another  curve,  called  the  herpolhode, 
which  is  fixed  in  space.  The  cones  projecting  these  curves 
from  0  are  known  as  Poinsofs  rolling  cones,  the  polhodal 
cono  rolling  over  the  fixed  hcrpolhodal  cone. 

438.  The  equations  of  the  polhode  as  the  locus  of  those 
points  of  the  momental  ellipsoid  whose  tangent  plane  has  the 

22 


322  KINETICS  [439. 

constant  distance  8  from  0  are  evidently 

/i.Tr  +  hrji'  +hzi'  =  1,     h'x^'  +  hV  +  hW  =  ~T' 

0" 

?'.  e.  the  polhode  is  the  intersection  of  the  momenta  1  ellipsoid 
with  a  coaxial  elUpsoid.  Multiplying  the  second  equation 
by  5^  and  sul)tracting  the  rei-ult  from  tlie  first  equation  we 
ol)tain  the  equation  of  the  -poniodal  cone 

7i(l  -  /i5-).fi2  +  /o(l  -  /o5-')7/i2  +  73(1  -  735^-)2i2  =  0. 

If  we  take  the  notation  so  that  /i  >  72  >  73  this  cone  is 
real  if  and  only  if 

For  5"  =  1/73,  the  polhode  reduces  to  a  point,  viz.  the  ex- 
tremity of  the  longest  axis  of  the  momental  ellipsoid.  As  5^ 
diminishes,  the  polhode  is  first  an  oval  about  this  longest  axis. 
When  5-  =  1/72,  the  polhoclal  cone  degenerates  into  a  pair  of 
planes  and  the  polhode  l^ecomes  an  ellipse.  When  5^  lies 
between  1/72  and  l/7i  the  polhode  is  an  oval  about  the 
shortest  axis,  and  it  contracts  to  the  extremity  of  this  axis 
for  52  =  1//,. 

For  values  of  5^  very  close  to  1/72  the  motion  can,  in  a 
certain  sense,  be  called  unstable  since  a  slight  disturbance 
might  change  the  polhodal  cone  from  a  cone  about  the  longest 
to  a  cone  about  the  shortest  axis,  or  vice  versa. 

439.  The  herpolhode  is  a  plane  curve;  but  it  is  in  general 
not  closed.  The  radius  vector  OP  =  p  (Fig.  88),  if  not  con- 
stant, has  a  greatest  and  a  least  value  in  the  course  of  the 
motion,  and  the  same  is  true  of  its  projection  QP  on  the 
fixed  plane.  Hence  the  herpolhode  lies  between  two  con- 
centric circles.     When  p  is  constant  these  circles  coincide 


441.]  RIGID  BODY  WITH  A  FIXED  POINT  323 

and  the  herpolhode  coincides  with  them.  It  can  be  shown 
that  the  herpolhode  has  no  points  of  inflection. 

440.  The  invariable  line  describes  a  cone  in  the  moving 
body.  Its  equation  may  be  found  from  the  reciprocal 
ellipsoid 

^  ,y^  i^A  =  ^ 

whose  radius  vector  in  the  direction  5  is  1/5  (Arts.  398,  399), 
and  hence  constant.  The  cone  must  pass  through  the  inter- 
section of  the  reciprocal  ellipsoid  and  the  sphere 


0" 


Hence  its  equation  is 


h)  ■''-+{''- h) '■"'-  +  {'"- b '-'- '■ 


441.  When  H  =  0  Eider's    equations  (11),  Art.  432,  are 

/iCOi    =    (/o    —    l2)W2W-i,        I'i<^2    =    (h    —    /OcOgOJl, 
ho:-6   =    (/i   —   /2)C01W2. 

Multiplying  by  coi,  coo,  ws  and  adding  we  find 

~  i(/ia;:2  + /2coo2  +  /3C032)  =  Q; 

hence,  by  (9),  Art.  429, 

/icoi2  +  1,0,/  +  I,o:./~  =  2T  =  const.  (14) 

This  is  the  integral  of  kinetic  energy  and  work. 

Multiplying  (13)  by  /iwi,  /2CO2,  ho^z  and  adding  we  find 
similarly  Ijy  (3) : 

/rwr  +  /2"a32"  +  L^ws^  =  li^  =  const.,  (15) 

which  is  the  integral  of  angular  momentum. 


324  KINETICS  [442. 

As,  moreover, 

wi^  +  ^-i"  +  ^z'  =  w^,  (16) 

we  can  solve  (14),  (15),  (16)  for  coi^,  t02-,  wa^.     Introducing  the 
new  constants  a,  /3,  7  by  putting 

2T{l2  +  73)  -h^  =  hha\     2T{h  +  7i)  -h^  =  IzL^\ 

2T{h  +  h)  -  h^  =  Iihy\ 


we  find 


"^'  =  (/i  -  mh  -  h)  ^"' "  "'^' 


^^^1  (^2_„2)^  (17) 


(/2    -    /3)(/l   -   U) 


o  7i/2  .      2  ON 

-3-    =     (7,    _    /3)(/2    _    73)    (-      -    T^)- 

Hence,  if  Ji  >  /o  >  /a  we  have  w^  >  a^,  co^  <  iS^,  co^  >  72. 

442.  To  find  the  time,  multiply  the  equations  (13)  by  coi//i, 
W2//2,  (jiz.Hz  and  add: 

(/l    -    /2)(/l    -    /3)(/2    -    /s) 

«(-2W  )    =    7—}^^: W1CO2CO3; 

-1  li2^3 

substituting  for  coi,  0)9,  ws  their  values  (17)  we  find: 


V     V(C02    -    «2)(^2    _    ^2)  (^2    _    ^2) 


The  positive  or  negative  sign  must  be  used  according  as  d{w^) 
is  positive  or  negative. 

As  t  is  given  by  an  elhptic  integral,  co^  is  a  periodic  function 
of  the  time. 

443.  If,  in  particular,  the  momental  ellipsoid  at  0  is  an 
ellipsoid  of  revolution,  say  if  /i  =  lo,  the  results  assume  a  very 
simple  form.     Euler's  equations  (13)  reduce  to 

Wi    =    XcOoWS,        ^2    =     —    XcOsCOi,        CO3    =    0,  (18) 


444.]  RIGID  BODY  WITH  A  FIXED  POINT  325 

where 

The  angular  velocity  cos  about  the  third  axis  Ozi  (which  is  not 
necessarily  an  axis  of  symmetry  for  the  mass  of  the  whole 
body)  is  therefore  constant: 

C03  =  n. 

The  first  two  equations  (18)  give  coiwi  +  C02W2  =  0,  whence 

toi^  +  co2^  =  const.  =  rn}. 
It  follows  that 


CO    =    l/cor   +   0)2^   +   COs^    =    Vm-   +   71^ 

is  constant  although  coi  and  C02  vary. 

The  inclination  of  the  instantaneous  axis  to  the  principal 
axes  Oxi,  Oyi  varies,  but  its  inclination  to  the  third  principal 
axis  Ozi  is  constant,  viz.  cos~^(co.-i/a)).  This  means  that  the 
polhodal  cone  is  a  cone  of  revolution  about  Ozi  and  the 
polhode  is  a  circle.  The  herpolhode  is  therefore  Hkewise  a 
circle  (Art.  439).  As  the  two  circular  cones  are  in  contact 
along  the  instantaneous  axis,  this  axis  lies  in  the  same  plane 
with  the  impulse  h  and  the  axis  Ozi. 

444.  To  find  coi,  0^2  separately,  differentiate  the  first  equa- 
tion (18)  with  respect  to  t  and  substitute  for  cj^  its  value  from 

the  second: 

0)1  +  X^n^coi  =  0; 
hence 

Oil  =  k  sin(Xn^  +  e), 

where  k,  e  are  the  constants  of  integration.     The  fir^  equa- 
tion (18)  then  gives 

012  =  z-  Oil  =  k  cos(Xnf  +  e). 
Kn 


326  KINETICS 


1445. 


As    coi^  +  C02-  =  7)1^    (Art.    443)    it     appears     that    k  =  m. 
Hence 

coi  =  7nsin(\nt  +  e),     C02  =  wz  cos(\nt  +  e),     C03  =  ?i.    (19) 

445.  To  determine  the  position  of  the  body  with  respect 
to  fixed  axes  through  0  let  the  invarialilc  direction  of  h  be 
taken  as  axis  Oz.  The  direction  cosines  of  h  given  in  Art. 
435  give 

hi  =  1 10)1  =  h  smO  sinv?,     ho=  12(^2=  h  sin0  cos^, 

hs  =  hooi  =  h  cos^. 
It  follows  that 

COS0  =  ~T-   =  const.,     tan^  =  —  =  tan(XwY.  +  e): 

hence  <p  =  \nt  +  e  and  6  =  0,  (p  =  \n  =  const. 

Finally,  the  third  of  the  equations  (12),  Art.  435,  gives 

,  _n  -\n  _  (1  -  \)h  _  Ji 
'^  co"s^  ~        h       ~  Ii' 

whence  x}/  =  (h'Ii)t  +  \po- 

Thus  if  we  resolve  w  along  the  oblique  axes  ON,  OZi,  OZ 
(Art.  435)  into  d,  <p,  yp  (see  Fig.  87),  we  have  (?  =  0  while  <p  and 
^  are  constant.  The  motion  of  the  body  consists  therefore 
in  the  rotation  of  constant  angular  velocity  <p  =  \n  about 
OZi,  together  witli  the  turning  of  this  axis  OZi  with  constant 
angular  velocity  \}/  =  hITi  about  the  axis  OZ,  the  angle  6  = 
ZOZi  between  these  axes  remaining  constant.  Such  a  motion 
is  called  a  regular  precession;  the  nodal  hne  OA^  (Fig.  87) 
is  said  to  precess  with  the  velocity  of  -precession  ^;  OZ  is  the 
axis  of  precession. 

If,  in  particular,  the  momental  ellipsoid  at  0  is  a  sphere, 
so  that  1 1  =  I2  and  hence  X  =  0,  we  have  ^  =  0;  hence  the 


446.]  RIGID   BODY  ¥/ITH  A  FIXED  POINT  327 

whole  motion  consists  of  the  rotation  of  angular  velocity  xp 
about  the  fixed  axis  OZ.  This  was  to  be  expected;  for,  as  a 
principal  axis,  OZ  is  a  permanent  axis  of  rotation  (Art.  424). 

3.  Heavy  symmetric  top. 

446.  A  rigid  body  with  a  fixed  point  is  often  spoken  of  as 
a  top  although  the  ordinary  children's  top  has  no  fixed  point 
but  has  merely  one  of  its  points  approximately  confined  to  a 
plane  or  other  surface. 

If  the  momental  ellipsoid  at  the  fixed  point  0  is  an  ellipsoid 
of  revolution,  say  about  Ozi,  so  that  7i  =  h,  and  the  centroid 
G  of  the  body  lies  on  Ozi,  say  at  the  distance  OG  =  k  from  0, 
the  body  is  called  a  symmetric  top.  If,  moreover,  the  only 
force  acting  on  the  body  (besides  the  reaction  at  0)  is  the 
weight  W  of  the  body  we  have  the  heavy  symmetric  top. 

If  k  were  zero  we  should  have  the  case  of  Arts.  443-445. 
If  /c  4=  0  but  the  initial  angular  velocity  be  zero,  the  body 
would  swing  like  a  compound  pendulum  in  a  vertical  plane. 
With  proper  initial  conditions  the  heavy  symmetric  top  may 
move  like  a  (compound)  spherical  pendulum  with  Ii  =  h 
at  0.  But  in  speaking  of  the  motion  of  the  heavy  symmetric 
top  it  is  generally  understood  that  the  initial  angular  velocity 
is  large  and  takes  place  al)out  an  axis  not  differing  very  much 
from  the  axis  Ozy.  To  explain  what  is  here  meant  by  large 
observe  that  if  in  the  course  of  the  motion  the  centroid  G 
rises  or  descends  through  a  vertical  distance  z  the  work  of 
gravity,  ±  Wz,  changes  the  kinetic  energy  of  the  top. 
Now  this  variation  in  the  kinetic  energy  can  never  amount  to 
more  than  2Wk.  Hence  if  k  is  reasonably  small  and  the 
initial  angular  velocity  large,  the  initial  kinetic  energy  will  not 
he  affected  very  much  by  the  changes  due  to  the  rise  and  fall 
of  the  centroid  G.     It  is  especially  cases  of  this  kind  that  we 


328  KINETICS  [447. 

have  in  mind  when  speaking  of  the  phenomena  of  the  top. 
The  general  equations  of  Arts.  447,  448,  however,  do  not 
imply  any  such  restricting  assumptions. 

447.  Taking  the  fixed  axis  Oz  vertical  and  positive  up- 
ward and  the  moving  axis  Ozi  along  the  third  principal  axis 
at  0,  we  find  Euler's  equations  (11)  in  the  form 

Zicoi  +  {h  —  /i)w2W3  =  Wk  smd  coscp, 
Iiuio  +  (/i  —  Izjoiiuz  =  —  Wk  sin0  sin<p, 
/sojs  =  0,     C03  =  const.  =  n. 

The  integral  of  kinetic  energy  and  work  is 

/icoi^  +  /icoo2  _^  73^32  =  2Wk{coQdo  -  COS0)  4-  2 To, 

Gq  and  To  being  the  initial  values  of  the  angle  zOzi  =  6  and 
the  kinetic  energy  T. 

The  angular  momentum  about  the  axis  Oz  being  constant 
we  have 

7icoi  sin9  sincp  +  7,co2  sin0  cos^  +  I^n  cos0  =  const.  =  hz. 

If  coi  and  C02  be  replaced  by  their  values  (12),  Arts.  435,  the 
two  first  integrals  l^ecome 

7i(^2  _^  ^2pij-^20)  =  2WA-(cos0o  -  cos9)  -  73n2  +  27^0, 
Iii/  sin-0  =  —  73W  COS0  +  h/, 

eliminating  i/-  we  have  for  the  determination  of  6: 

IP  =  2TT^A:(cos0o  -  cos0)  -  73^2  +  27^0  -  ^JhJZ^'^^^Jl  ^ 

1 1  sm  u 

or  introducing  cos9  =  u  as  new  variable : 

Having  found  u  from  this  equation  we  have  for  \l/ : 


449.]  RIGID  BODY  WITH  A  FIXED  POINT  329 

.     _    h^  —    IzTlU 

and  then  (p  can  be  found  from  the  third  equation  (12)  which 

gives 

1   hz  —  hnu 

448.  To  discuss  the  equation  for  u  let  us  put 

/i[2Tf/b(Mo  -  w)  +  27^0  -  Iin'\{l  -  ii~)  -  {h  -  h^iuy  = /{u) 
so  that 

Iiu  =  ±  ^lf{u). 

As  /(  —  1)  <  0,  /('Wo)  >  0  (because  initially  u  is  real), 
/(I)  <  0,  /(oo)  >  0,  the  cubic  /(w)  has  three  real  roots, 
say  Ui,  U2,  Us,  such  that 

—  I  <  Ui  <  Uq  <  ih  <  '^  <  Ua  <  oo. 
For  the  time  we  have 


^  -  "  y^wk  j 


du 


V(w  —  Ui){u  —  Uo){u  —   Us) 

the  plus  or  minus  sign  being  used  according  as  du  is  positive 
or  negative.  As  u  =  cos0  must  lie  between  —  1  and  +  1 
it  oscillates  between  its  least  value  Ui  and  its  greatest  value 
u^;  i.  e.  the  axis  Ozi  oscillates  between  its  greatest  inclina- 
tion di  and  its  least  inclination  do  to  Oz. 

449.  Suppose,  in  particular,  that  the  body  is  initially 
given  a  spin  about  the  third  principal  axis  OZi  so  that 
wi  =  0,  C02  =  0  for  t  =  0.  We  may  take  the  axes  of  refer- 
ence so  that  (p  =  0  and  ^i-  =  0  for  ^  =  0.  We  then  have 
since  hz  is  constant : 

hz  =  hn  COS0O, 
and 

f{u)  =  (wo  -  u)[2IiWk(l  -  u'-)  -  /3-n2(^/,o  -  w)]. 


330  KINETICS  [450. 

When  Uo  <  u  <  1,  f{u)  is  clearly  negative;  it  is  therefore 
U2  which  is  equal  to  Uo-  Hence,  at  the  beginning  of  the 
motion  u  diminishes;  in  other  words,  do  is  the  minimum 
inclination  of  the  axis  OZi  to  the  vertical  OZ. 

450.  The  centroid  G  descril^es  a  spherical  curve;  its  pro- 
jection on  the  horizontal  A"F-plane  lies  between  the  circles 
of  radii  k^il  —  Wi^  and  k-yjl  —  u^^  about  0.  The  co-ordinates 
X,  y  of  the  projection  of  the  centroid  on  the  A^F-plane  are 


X  =  k^ll  —  u^  sim^,      y  =  —  k^l  —  u^  cos^. 

To  determine  the  direction  in  which  the  curve  approaches 
the  bounding  circles  let  us  determine  the  angle  /x  between  the 
radius  vector  p  and  the  tangent  to  the  curve.     We  have 

p  Vl  —  U"  1  —  u^  d^p 

tan  u.  =  -,—  =  =t =  =F J- . 

^P  d     r 5  u       du 

Now  by  Arts.  447  and  449 

d\p  .       I371U0  —  u 

du  Ii  1  —  u^  ' 

hence 

,  hnuo  —  u 

tan  u  =  =F  ~r^ : — . 

il        uu 


As  Iiu  =  ±  V/(w)  (Art.  448)  we  find 


,  ^     Uo  —  u  IzU  Vwo  —  u 

tan  n  =  Izii 


u  V/(m)      u  ^j2WkIi{u  —  Wi) {u  —  Us) 


This  shows  that  tan  fi  becomes  infinite  for  u  =  Ui  and  zero 
for  ti  =  Uo  =  u^.  The  curve  meets  therefore  the  inner 
circle  at  right  angles  (with  a  cusp)  and  touches  the  outer 
bounding  circle.     It  is  in  general  not  a  closed  curve. 


452.)  RIGID  BODY  WITH  A  FIXED  POINT  331 

451.  The  expressions  for  9,  ip,  \p  as  functions  of  i  assume 
a  simple  form  if  we  suppose  the  initial  angular  velocity  n 
about  OZi  to  be  very  large  (Art.  446).  In  this  case  the 
equation  (Art.  449) 

fiu)  =  (mo  -  u)[2IiWkil  -  u'~)  -  Mi-(wo  -  w)]  =  0 

has  its  root  Wi  nearly  equal  to  Wo  so  that  the  angle  6  differs 
but  little  from  ^o-  Hence  if  we  put  6  =  do  -\-  v,  v  will  be 
small.     This  gives  cos9  =  cos^o  —  v  sin^o,  i-  e. 

COS0O   —   COS0        .     .  •     ra       \       \  •     n       \  a 

V  = .--- ,  sm0  =  sm(»o  -r  v)  =  sm^o  +  v  cos^o- 

smpo 

Substituting  these  values  in  the  equation  for  6  (Art.  447) 
we  find 

Zi^^^  =  2Wkhv  sin^o  -  h-if^        l'T'^'\.^, 

(sm0o  +  V  cos^o) 

or  neglecting  the  term  v  cos^o  in  comparison  with  sin^o: 

lid  =  ^I2WkIlV  sindo  -  hVv^ 

As  6  =  V  we  find  upon  integration 

7i  •     ,v  ,  Wkli  sin^o 

t  =  -^ —  versm  ^  -  ,     where  a  = zr-r—;: — , 

Un  a  U^n^ 

and  hence 
6  =  Oo  +  V  =  00  -^  a  (l  -  cos  -p  A  =  eo-\-2a  sin^  ^  t. 

The  variation  v  in  the  value  of  9  is  called  the  nutation; 
it  is  periodic,  of  period  2TrIi/hn. 

452.  By  Arts.  447  and  449, 

»  _  hn  cos^o  —  COS0  _  IsU     j'_ 
/i         sin^^  1 1  sin^o ' 

where  (Art.  451) 


332  KINETICS  [453. 

p  =  a  {    1  —  cos  ^^  t 

Hence,  integrating  and  observing  that  \{/  =  0  ior  t  =  0: 

,  hna    ,  a      .    I^n  ^ 

iism^o        sm^o        /i 

Thus  the  first  term  of  \p  increases  uniformly  with  the  time, 
while  the  second  is  periodic.  The  angular  velocity  \l/  is  the 
velocity  of  precession  (Art.  445). 

453.  For  <p  we  have  by  Art.  447: 

hncosOo  —  eosd     ..  1 371 

(p  =  n ,—  ■ ,-— cot5  =  n :^v  cot^o 

1 1  svnd  1 1 

/sn  /  Iz7i 

=  n  —  J  -cot^o  ■  ('  \\  —  cos-Y^t 

hence 

(p  =  in  —    j~a  cot^u  \t  -\-  a  cot^o  sin— ^—  t. 

454.  Let  us  finally  inquire  into  the  conditions  under  which 
the  top  while  spinning  about  its  axis  OZi  may  keep  its  inclina- 
tion d  =  ZOZi  to  the  vertical  constant.  A  motion  of  this 
kind  is  often  spoken  of  as  stable,  or  steady. 

As  6  is  to  be  constant  we  find  from  Art.  447  that  the  velocity 

of  precession, 

hz  —  hn  cos9 

\j/  = • 

1 1  sin-d 

remains  constant,  say  =  i/'o;  and  similarl}'  the  velocity 

<p  =  n  —  \p  cos9 

remains  constant,  say  =  <po.  The  motion  is  therefore  a 
regular  precession  (Art.  445). 


454.] 


RIGID   BODY  WITH  A  FIXED  POINT 


333 


The  angular  velocity  co,  at  any  instant,  has  the  components 
<Po  along  OZi  (Fig.  89)  and  i/'o  along  OZ;  let  us  resolve  it  along 
OZi  and  the  perpendicular  OPi  to  OZi  in  the  plane  ZOZi; 
the  components  will  be  ^o  +  '/'o  cosO  along  OZi  and  rpo  smd 


Fig.  89. 

along  OPi.  As  the  moment  of  inertia  about  OZi  is  h  and 
that  about  any  perpendicular  to  OZi  is  /i,  the  angular  mo- 
mentum about  OZi  is  h{<PQ  +  'Ao  cos0),  while  that  about  OPi 
is  /iiAo  sin0.  Hence  the  angular  momenta  about  OZ  and  the 
perpendicular  OP  to  OZ  in  the  plane  ZOZi  are 

hi^Po  +  '/'o  COS0)  COS0  +  /i^Z-o  sin2(?, 
-^^3(^0  +  ^0  cos^)  sin0  —  /I'/'o  sin^  cos0. 

The  former  component  is  constant;  the  latter,  about  OP, 
receives  in  the  element  of  time  the  increment 

[h(<Po  +  ^0  COS0)  smd  —  Ii\po  m\d  cos(9](i/'o  +  <Po  cosd)df. 

If  the  motion  is  to  be  steady  this  increment  must  just 
equal  the  angular  momentum  about  OP  imparted  to  the 
body  by  the  force  of  gravity  in  the  time  element,  i.  e.  to 
Wk  sin0  dt.     Hence  the  condition 

[73(<^o+'/'o  COS0)  fi\nd—  I li/o  smO  cosd](\j/o-{- (fo  cosd)  =  Wk  smd. 


334  KINETICS  [454. 

This  requires  either  sin0  =  0  which  would  mean  that  the 
axis  of  the  top  is  vertical,  or 

1/3(^0  +  '/'o  coiid)  —  Ii\po  cos^KiAo  +  (Pq  COS0)  =  Wk. 

For  given  values  of  <po  and  i/o  this  condition  can  in  general 
be  satisfied  ])y  two  different  values  of  cos0  since  the  equation 
is  quadratic  in  cos0. 

For  a  further  study  of  the  motion  of  tops  and  gyroscopes 
the  following  works  may  be  consulted:  H.  Crabtree,  An 
elementary  treatment  of  the  theory  of  spinning  tops  and 
gyroscopic  motion,  London,  Longmans,  1909;  A.  G.  Webster, 
The  dynamics  of  particles,  etc.,  Leipzig,  Teubner,  1904; 
F.  Klein  und  A.  Sommerfeld,  L^eber  die  Theorie  des  Kreisels, 
Leipzig,  Teubner,  1897-1910. 


CHAPTER  XIX. 
RELATIVE   MOTION. 

455.  We  shall  here  consider  only  the  motion  of  a  particle 
relatively  to  a  rigid  body  B  having  a  given  motion  with 
respect  to  fixed  axes.  By  the  theorem  of  Coriolis  (Art.  150), 
the  absolute  acceleration  j  of  the  particle  is  the  resultant  of 
the  body  acceleration  jb,  the  complementary  acceleration 
jc  =  2coVr  cos(a;,  Vr),  and  the  relative  acceleration  jV: 

J  =  jb  +  jc  +  jr. 

If  771  is  the  mass  of  the  particle,  F  the  resultant  of  the  given 
forces  acting  upon  it,  its  equation  of  motion  is  7nj  —  F. 
Hence,  multiplying  the  equation  of  Coriolis  by  m  and  putting 

—  mjb  =  Fb,     —  mjc  =  Fc, 
we  find 

mjr  =  F  +  Fb  -;-  F,. 

This  vector  equation  gives  by  projection  on  the  moving  axes 
OiXi,  Oiiji,  OiZi,  rigidly  connected  with  the  body  of  reference 
B: 

mxi  =  X  +  Xb  +  Xc, 

mij^  =  F  +  Yb  +  Yc,  (1) 

m'zi  =  Z  +  Zb  -\-  Zc. 

Here  X,  Y,  Z  are  the  components,  along  the  moving 
axes,  of  the  resultant  F  of  all  the  given  forces  acting  on  the 
particle.  Xh  Yb,  Zb,  are  the  components,  along  tiic  same 
axes,  of  Fb  =  —  mib,  where  m  is  the  mass  of  the  particle  and 
^6  the  acceleration  of  that  point  of  the  body  B  with  which 

335 


33G  KINETICS  [456. 

the  particle  happens  to  coincide  at  the  instant  considered; 
Fb  may  be  called  the  body-force.  Xc,  Yc,  Zc  are  the  com- 
ponents of  the  complementary  force  Fc  —  —  mjc,  where  jc  is 
a  vector  of  length  2coVr  sin(a;,  v^),  at  right  angles  both  to  the 
rotor  CO  of  the  body  B  and  to  the  relative  velocity  Vr  of  the 
particle  with  respect  to  B. 

Hence  we  may  say  that  the  equations  of  the  relative  motion 
of  the  particle  m,  i.  e.  of  its  motion  as  it  would  appear  to  an 
observer  moving  with  the  body  of  reference  B,  are  formed 
like  the  equations  of  absolute  motio7i,  except  that  to  the  given 
forces  acting  on  the  particle  must  be  added  the  body-force  and 
the  complementary  force. 

456.  It  may  be  noted  that  the  body-force  Fb  =  —  mjb 
vanishes  only  when  the  point  of  B  with  which  the  particle 
coincides  moves  uniformly  in  a  straight  line,  and  that 
Fc  =  —  2mwVr  sin(co,  Vr)  vanishes: 

(a)  when  w  =  0,  i.  e.  when  the  body  B  has  a  motion  of 
translation; 

(6)  when  Vr  =  0,  i.  c.  when  the  particle  is  in  relative  rest; 

(c)  when  sinfoj,  y,)  =  0,  i.  e.  when  the  relative  velocity  Vr 
of  the  particle  is  parallel  to  the  rotor  co,  i.  e.  to  the  instan- 
taneous axis  of  B. 

The  principle  of  kinetic  energy  and  work  gives 
hnvy^-hnv,''  =  rUX  +  Xb)dx,  +  {Y  +  Yb)dy,  +  {Z+Zb)dzr] 
since  the  work  of  the  complementary  force  Fc  which  by 
definition  is  normal  to  the  velocity  iv  is  always  zero. 

457.  Motion  and  rest  relatively  to  a  body  B  rotating  uniformly  about 
a  fixed  axis. 

If  P  (Fig.  90)  be  that  point  of  B  at  which  the  particle  m  is  situated 
at  the  time  t,  OP  =  r  its  distance  from  the  fixed  axis  (through  0),  the 
acceleration  of  P  is  jb  =  —  wV.  Hence  Ft  =  mwh  is  directed  along 
OP  away  from  the  axis;  i.  e.  the  body  force  Fj  is  in  this  case  what  is 
commonly  called  the  centrifugal  force. 


458. 


RELATIVE  MOTION 


337 


//,  in  particular,  the  particle  is  absolutely  at  rest,  its  relative  velocity 
Vr,  i.  e.  the  velocity  which  it  appears  to  have  to  an  observer  at  P  moving 
with  the  body  B,  is  equal  and  opposite  to  the  velocity  %  =  wr  of  the 
point  Poi  B.     As  regards  the  accelerations,  observe  that  jt  =  —  coV  and, 


\'^ 

w 

1 
0 

r 

4 

P 

f 

"  », 

Fig.  90. 


since  ^(w,  Vr)  =  \ic,  jc  =  2o3Vr  =  2co-r.     The  sense  of  jc  is  away  from 

the  axis,  i.  e.  opposite  to  jb.     The  apparent  motion  of  the  particle  is 

a  uniform  rotation  about  the  fixed  axis  opposite  to  that  of  the  body; 

hence  the  relative  acceleration  is  jr  =  —  coV. 

The  absolute  acceleration  j  is  therefore  =  jr  + 

ji.  -\-  jc  =  —  o?r  —  wV+  2coV  =  0,  as  it  should 

be. 

458.  Motion  of  a  heavy  particle  m  on  a 
straight  line  turning  uniformly  about  a  vertical 
axis  whose  downward  direction  is  met  by  the  line 
at  a  constant  angle  a  <\-k  (Fig.  91). 

Taking  as  origin  the  intersection  0  of  the 
line  with  the  axis  we  have  Vr  =  r  where  r  = 
OP,  The  complementary  force  is  taken  up 
by  the  reaction  of  the  tube. 

The  components  of  the  weight  mg  and  of 
the  body-force  ynwh  sina  along  OP  are  7ng  cosa 
and  mwh  sin-a;  hence  the  equation  of  relative 
motion: 

f  =  g   cosa  +  co-r  sin^a 

Putting  r  -j-  g  cosa/oP  sin^a  =  u  we  have 


whence 


23 


il  =  (to  sina)-?/, 


,     0  COSor  ^    I 

M  =  r  +    „   .   ,     =  Tie' 


+  C,e 


-(a>  e\r\ay 


838 


KINETICS 


1459. 


If  the  particle  starts  from  rest  at  0  (or  rather  from  a  point  very 
near  to  0)  we  find 


hence 


2u)^  sin^a 


g  cosa 

1 2  - 


Ci  —  62  —  ^ 

Jf_C0Sa^  /„i„  smat 


silia-ty^ 


For  the  projection  of  the  path  on  the  horizontal  plane  we  have 
P  =  0  sinof,  6  =  cot;  hence  the  projection  of  the  absolute  path  on  the 
horizontal  plane  is 


which  represents  a  spiral. 

459.  Motion  of  a  -particle  relative  to  the  earth,  near  its  surface. 

The  earth's  motion  of  translation  (which  is  not  uniform)  need  not 
be  considered  since  the  forces  affecting  it  act  on  the  particle  just  as 


Fiff.  92. 


they  do  on   the   earth   and   hence  do  not  affect  the  relative  motion. 
The  earth  can  therefore   be   regarded  as  rotating  uniformly  about  a 
fixed  axis;  the  slight  variation  of  direction  of  the  axis  may  be  neglected. 
The  angular  velocity  of  the  earth  is 

27r 
"  =  ^a^c^T~^  =  0.000  072  92  rad./sec, 
86 164.1  ' 

the  sidereal  day  having  86  164.1  sec.  of  mean  time. 


459.] 


RELATIVE   MOTION 


339 


The  body-force  is  simply  the  centrifugal  force  (Art.  458)  mco^r  = 
mco^R  coS(^,  where  R  is  the  earth's  radius  and  <{>  the  latitude. 

In  most  problems  of  relative  motion  near  the  earth's  surface  the 
introduction  of  this  centrifugal  force  is  unnecessary.  This  is  best 
seen  by  considering  a  particle  at  relative  rest,  say  the  bob  of  a  pendulum 
hanging  at  rest  (Fig.  92).  Let  P  be  the  bob,  S  the  point  of  suspension, 
0  the  earth's  center,  OP  =  R  the  earth's  radius,  r  =  R  cos^  the  radius 
of  the  parallel  in  latitude  </>. 

As  Vr  =  0,  the  complementary  force  is  zero;  hence  the  only  forces 
to  be  considered  are  the  centrifugal  force  mu^r,  the  tension  of  the  rod 
along  PS,  and  the  earth's  attraction  which  is  directed  along  PO  if 


Fig.  93. 


we  regard  the  earth  as  composed  of  homogeneous  spherical  layers. 
Hence  the  tension  of  the  rod  must  balance  the  resultant  of  the  cen- 
trifugal force  and  the  attraction.  But  this  resultant  is  due  precisely 
to  the  actually  observed  acceleration  g  of  falling  bodies  since  this  in- 
cludes the  combined  effect  of  centrifugal  force  and  attraction. 

The  complementary  force,  —  2ma}Vr  sina,  where  a  is  the  angle  between 
the  relative  velocity  v,-  and  the  earth's  axis  (northward)  is  at  right  angles 
to  the  plane  of  the  angle  a.     We  take  the  earth's  center  O  as  origin  of 


340  KINETICS 


1460. 


the  fixed  axes  and  Oz  toward  the  north  (Fig.  93);  the  origin  of  the 
moving  axes  at  any  point  P  (in  latitude  0)  on  the  earth's  surface, 
Fzi  vertical,  Px\  tangent  to  the  meridian  southward,  and   hence  Py\ 
tangent  to  the  parallel  eastward. 
We  then  have:  . 

cji  =  CO  cos(7r  —  <^)  =  —  CO  cos<^,     C02  =  0,     cos  =  CO  sin<^. 

Hence  the  components  of  the  complementary  acceleration  jc  are 

2(co2ii  —  co3?/i)  =  —  2co2/i  sin0, 

2(co3.ri  —  coiii)  =  2oo(.ri  sine/)  +  Zi  cost^), 

2(a)i7/i  —  C02.ri)    =    —  2coi/i  COS0. 

The  components  of  the  comijlementary  force  Fc  along  the  moving  axes 

are  therefore: 

Xc  =  2mcoyi  sin<^, 

Yc  =  —  2mw{zi  cos</)  +  xi  sin0), 

Zc  =  2mco?/i  cos(j>. 

460.  Relative  ynotion  of  a  heavy  particle  on  a  smooth  horizontal  plane. 
The  centrifugal  force  being  taken  into  account  by  using  the  observed 
value  of  g  (Art.  461)  the  equations  of  the  relative  motion  are 

N 
ii  =  2co32/i,     2/1  =  2(coi2i  —  cos.ri),     Zi  =  --  —  g  -  2coi?/i, 

m 

where  N  is  the  normal  {i.  e.  vertical)  reaction  of  the  plane.  As  Zi 
and  ii  are  constantly  zero,  the  equations  reduce  to 

xi  =  2co3?/i,     yi  =  -  2CO3X1,     N  =  mig  +  2coi?/i), 

where  coi  =  —  co  cos</),  C03  =  co  s{n4>.  The  third  equation  determines  N 
as  soon  as  2/1  has  been  found  from  the  first  two.  The  principle  of 
kinetic  energy  and  work  gives 

iCii^  +  2/1^)  =  const. 

Hence  the  relative  or  apparent  velocity  Vr  is  constant. 

Assuming  the  particle  to  start  from  the  origin  P  we  find  by  integrating 
each  of  the  two  equations  by  itself: 

.fi  =  .To  +  2co32/i,      2/1=2/0—  2aj3Xi; 

as  -fr  +  2/1"  =  I'r^  =  I'u"  =  i'o"  +  2/0^  we  find  as  equation  of  the  path: 


461.1  RELATIVE   MOTION  341 

\  2a)3  J         V  2a;3  /  VScoj  / 

a  circle  tangent  to  the  initial  velocity  in  the  horizontal  plane.     The 
center  C  (Fig.  94)  lies  on  the  perpendicular  to  vo  tlirough  P,  to  the 


right  of  an  observer  at  P  looking  in  the  direction  of  vo,  in  the  northern 
hemisphere,  i.  e.  for  positive  (j>,  to  the  left  in  the  southern  hemisphere. 

Thus  the  particle  deidates  to  the  right  in  the  7iorthern,  to  the  left  in  the 
southern  hemisphere. 

The  radius  of  the  circle  is  very  large  since  w  is  very  small.  Thus, 
for  (p  =  30°  we  have  for  this  radius 


Vo 
2a;3 


=    "  =  13700  Ik 


461.  Particle  falling  from  rest  in  vacuo.     The  equations  of  motion  are 
the  same  as  in  Art.  400  except  that  N  =  0: 

.fi  =  20)3^/1,     7/1  =  2cjiii  —  2co3.ri,     zi  =  —  g  —  2myi. 

If  the  starting  point  be  taken  as  origin,  the  initial  conditions  are 

Xo  =  0,     ?/o  =  0,     zo  =  0,     i-o  =  0,     2/0  =  0,     2o  =  0; 

hence  the  first  integrals  are 

i-i  =  2a)3?/i,     7/1  =  -  20)3X1  +  2wiZi,     2i  =  —  gt  —  2wi?/i. 


342  KINETICS  [462. 

The  method  of  successive  approximations  gives  the  first  approxi- 
mation 

±1  =0,     2/1=0,     ii  =  —  gt, 
whence 

Xi  =0,     2/1=0,     2i  =  —  hgt^. 

Substituting  these  values  in  the  expressions  for  the  velocities  we  find 
the  second  approximation 

i'l  =  0,     7/1  =  go:  COS0  •  /-,     2i  =  —  gt, 
whence 

Xi  =0,     iji  =  Igw  cos(j>  •  t^,     Zi  =  —  \gl^. 

The  third  approximation  gives 

Xi  =  J^w^  COS0  sin</) .  /^,      ?/i  =  Jgrco  cos^  •  t^, 
Zi  =  —  IgC-  +  Jfifw-  cosV  ■  f4. 

These  formula  show  not  only  an  easterly,  but  also  a  southerly  deviation ; 
the  latter  is  however  proportional  to  co^  while  the  former  is  proportional 
to  CO.  The  last  value  for  z  shows  that  the  earth's  rotation  slightly 
diminishes  the  vertical  distance  fallen  through  in  a  given  time. 

462.  The  eastern  deviation  of  a  falling  body  and  the  deviation  to 
the  right  of  a  projectile  (in  the  northern  hemisphere)  would  furnish 
an  experimental  proof  of  the  rotation  of  the  earth  if  they  could  be 
clearly  observed.  Experiments  on  falling  bodies,  with  this  purpose  in 
view,  have  been  made  repeatedly  in  the  last  century  and  even  earlier; 
and  the  mean  results  of  certain  attempts  of  this  kind  are  often  quoted 
as  confirming  the  theory.  But  an  examination  of  the  individual  results 
shows  these  so  widely  discrepant  that  no  reliance  can  be  placed  on  their 
mean.  In  the  case  of  projectiles,  such  as  rifle  bullets,  the  phenomenon 
is  masked  completely  by  the  very  much  larger  deviation  arising  from 
the  rotation  of  the  projectile  and  the  resistance  of  the  air. 

For  this  reason  Foucault's  pendulum  experiment,  first  made  in 
1851,  and  since  often  repeated  with  good  success,  is  of  particular 
interest.  On  a  fixed  earth,  a  pendulum  set  swinging  in  a  vertical  plane 
would  continue  to  swing  in  the  same  plane;  on  the  rotating  earth,  the 
plane  in  which  the  pendulum  swings,  remaining  fixed  in  space,  must 
apparently,  i.  e.  relatively  to  the  earth,  turn  about  the  vertical  through 
the  point  of  suspension,  in  the  sense  opposite  to  that  of  the  earth's 
rotation,  with  the  angular  velocity  to  sin</),  where  co  is  the  angular  velocity 
of  the  earth  and  </>  the  latitude  of  the  place  of  observation. 


463.1  RELATIVE   MOTION  343 

463.  Foucault's  pendulum.  It  will  be  convenient  to  take  the  point 
of  suspension  0  as  origin,  the  axis  Ozi  vertically  downward,  Oxi 
tangent  to  the  meridian  northward,  and  hence  Oyi  tangent  to  the 
parallel  eastward.  The  forces  acting  on  the  bob  are  its  weight  mg, 
the  tension  N  of  the  suspending  wire,  and  the  complementary  force 
Fc  whose  components  are,  since  coi  =  co  cosqi,  C03  =  —  co  sin^ : 

Xc  =  —  2muiyi  siuij),    Yc  =  2??Jco(.ri  sin</>  +  Zi  cos4>),    Zc  =  —  2mco?/i  cos0. 


If  I  =  Vxi^  +  2/1^  +  2i^  is  the  length  of  the  wire  the  equations  of  motion 
are: 

Nxi      ^    .     . 
Xi  = r  ~  2,mh  smrf), 

Vi  = V  +  2coXi  sin0  +  2coZi  cos<A, 

7n  L 

Nzi       _    .        ^    , 

ml 

with  the  condition  P  =  .rr  +  yr  +  Zi^. 

The  general  integration  of  these  equations  would  present  serious 
difficulties.     But  for  small  oscillations  we  have 


\  P       )  2         V'  8  V        i^       / 


As  .Ti,  y\,  ii,  yi,  .fi,  iji,  are  small,  say  of  the  first  order,  zi  and  Zi  will  be 
small  of  the  second  order;  for  we  have  zr  =  P  —  xr  —  yi^,ZiZi  =  —  XiXi 
—  T/iT/i,  ZiZi  +  zi-  =  —  XiJt).  —  yiiji  —  Xi^  —  2/1^,  whence 

zi  =  -      (xixi  +  yiiji  +  ir  +  yi^  +  Zi-) 

Zi. 

= (xi.f  1  +  y\lh  +  .fi^  +  2/1=) r  {xih  +  yi2/i)*. 

Zi  2i' 

We  take  therefore  as  first  approximation  zi  =  0,  Zi  =  Z  so  that  the  third 
equation  of  motion  reduces  to 

A''  =  m{g  —  2a)?/i  cos</)). 

Substituting  this  value  in  the  first  two  equations  and  neglecting  terms 
of  the  second  order  we  find  if  we  write  w'  for  w  sin0: 

X,  +  2co'2/,  +\x,=  0,      7/,  -  2a;'.r,  +  'J  7/1  =  0. 


344  KINETICS  l464. 

Multiplying  by  yi,  Xi  and  subtracting  we  have 

Xiyi  —  yiXi  =  2co'(xi.ri  +  2/12/1), 
that  is: 

^^  (a:i2/i  -  2/1X1)  =  w'  ^^  (zr  +  2/i^). 

Hence,  integrating  and  putting  Xi  =  r  cosO,  yi  =  r  sinfl: 

r'~d  =  co'r=  +  C. 

If  r  =  0,  d  =  0  for  «  =  0  we  have  C  =  0  so  that 

0=  co', 
and  hence 

e  =  do  +  co'L 

This  means  that  the  apparent  motion  consists  of  the  rotation  of 
the  plane  in  which  the  pendulum  swings  about  the  vertical  with  the 
constant  angular  velocity  co'  =  co  sin0.  The  plane  makes  one  complete 
revolution  in  the  time  T  =  27r/a)  sin</). 

464.  In  theoretical  mechanics  the  motion  of  any  particle,  rigid 
body,  or  variable  system  is  referred  ultimately  to  a  reference  sj'stem 
(co-ordinate  trihedral)  which  is  regarded  as  fixed.  In  applying  mechanics 
to  the  study  of  physical  phenomena  we  meet  with  the  difficulty  that  in 
nature  no  absolutely  fixed  object  is  to  be  found.  For  motions  in  the 
vicinity  of  any  particular  point  on  the  earth's  surface  we  regard  the 
earth  as  fixed.  In  astronomy,  the  motions  of  the  planets  are  referred 
to  the  sun  as  if  it  were  a  fixed  center;  and  the  motion  of  the  solar 
system  is  referred  to  the  fixed  stars.  But  it  is  well  known  that  even 
the  so-called  fixed  stars  have  their  proper  motions.  Thus  in  all  these 
cases  we  are  merely  dealing  with  relative  motions. 

465.  It  should  be  observed  that  the  differential  equations  of  motion 
of  a  particle  are  the  same  whether  the  reference  system  is  at  rest  or 
has  a  rectilinear  uniform  translation.  In  other  words,  these  differential 
equations  admit  such  a  translation.  For,  if  for  x,  y,  z  we  substitute 
xi  -f  Vit,  2/1  +  vit,  Z\  +  vzt,  where  i'\,  V2,  Vi  are  the  constant  components 
of  the  velocity  of  translation,  we  have  x  =  X\,  ij  =  iji,  z  =  'z\. 

466.  Other  difficulties  in  the  fundamental  concepts  of  mechanics 
concern  the  idea  of  time. 

All  our  measurements  of  time  are  based  ultimately  on  the  assumption 
that  the  earth's  rotation  is  strictly  uniform.     That  this  assumption, 


466.1 


RELATIVE   MOTION  345 


which  can  not  be  verified  directly,  must  be  true  to  a  very  high  degree 
of  approximation  inay  be  inferred  from  the  agreement  of  astronomical 
predictions  with  actual  occurrences. 

Another  question,  and  one  that  has  been  much  discussed  in  recent 
years,  arises  from  the  difficulty  of  defining  the  simultaneity  of  two 
events  occurring  at  places  in  motion  relatively  to  the  observer  or 
observed  by  persons  in  motion  relative  to  each  other.  Consider  an 
observer  at  P  at  different  distances  from  the  points  A,  B.  If  the  times 
it  takes  light  to  travel  the  distances  AP  and  BP  are  h  and  ti,  then 
flashes  of  light  occurring  simultaneously  at  A  and  B  will  appear  to  the 
observer  to  happen  at  different  times,  the  difference  being  \ii  —  h\. 

Again,  if  the  observer  is  in  motion,  e.  g.  moving  toward  A  with 
velocity  v,  a  flash  given  at  A  when  the  observer  is  at  P  will  appear  to 
him  to  happen  at  a  time  AP/{V  +  v)  after  it  actually  occurred  (F  being 
the  velocity  of  light).  Thus  the  statement  that  two  events  are  simul- 
taneous does  not  have  a  definite  meaning  unless  the  position  and  motion 
of  the  observer  are  known. 

In  mechanics  we  deal  ordinarily  wdth  velocities  which  are  very  small 
in  comparison  with  the  velocity  of  light.  By  regarding  the  velocity 
of  light  as  infinite,  the  difficulty  would  disappear.  In  the  electron 
theory  where  the  moving  electron  has  a  velocity  comparable  with  that 
of  light  the  idea  becomes  of  importance. 


CHAPTER  XX. 

MOTION  OF  A  SYSTEM  OF  PARTICLES. 

I.  Free  system. 

467.  A  system  consisting  of  any  finite  number  of  particles 
is  called  free  if  the  co-ordinates  of  the  particles  are  subject 
to  no  conditions,  whether  these  be  expressed  by  equations 
or  inequalities.  The  forces  acting  on  any  one  of  the  particles 
are  distinguished  as  internal  or  external  according  as  they  are 
exerted  by  the  other  particles  of  the  system  or  proceed  from 
sources  outside'  of  the  system. 

Examples  of  such  systems  we  have  on  the  one  hand  in 
celestial  mechanics,  the  most  simple  case  being  the  problem  of 
two  bodies  (Arts.  321-327),  on  the  other  in  the  kinetic  theory 
of  gases  where  the  particles  are  the  molecules  of  the  gas. 

468.  Let  X,  Y,  Z  be  the  rectangular  components  of  the 
resultant  of  all  the  external  and  internal  forces  acting  on  any 
one  of  the  n  particles;  m  the  mass  and  x,  y,  z  the  co-ordinates 

of  the  particle;  then  the  equations  of  motion  of  this  particle 

are 

mx  =  X,     mij  =  Y,     mz  =  Z.  (1) 

There  are  3  such  equations  for  each  particle  and  hence  3n 
for  the  whole  system;  they  express  the  equilibrium  of  the 
external,  internal,  and  reversed  effective  forces. 

If  we  assume  that  the  internal  forces  occur  only  in  pairs 
of  equal  and  opposite  forces  between  the  particles,  depending 
only  on  the  mutual  positions  and  not  on  the  velocities  of  the 
particles,  almost  all  the  results  developed  in  Chapter  XV  for 

346 


469.]  MOTION  OF  A  SYSTEM  OF  PARTICLES  347 

the  system  of  particles  constituting  a  rigid  body  will  hold 
for  the  free  system,  except  that  we  have  now  3n,  instead  of 
merely  six,  equations. 

469.  D'Alombert's  principle  is  expressed  by  the  equation 

2(-wi;+  X)  8x-\-'^{-viy-\-  Y)  5?/+2(-m2+Z)  52  =  0,    (2) 

in  which  8x,  by,  8z  are  the  components  of  an  arbitrary  dis- 
placement 8s  of  the  particle  m.     As  the  3/1  virtual  displace- 
ments are  independent  of  each  other  this  equation  (2)  is 
equivalent  to  the  Sn  equations  (1). 
If  this  equation  be  written  in  the  form 

2m(x8x  +  ij8y  +  z8z)  =  S(X5a;  +  Y8y  +  Z8z)       (2') 

the  right-hand  member  will  contain  only  the  external  forces 
owing  to  the  assumption  (Art.  468)  concerning  the  internal 
forces. 

As  there  are  no  constraints  or  conditions  we  may  select 
for  8s  the  actual  displacement  ds  of  every  particle;  the  equa- 
tion 

'Lm{xdx  +  ydy  +  zdz)  =  2(Xrfrc  -f  Ydy  -{-  Zdz) 

then  gives  upon  integration  the  equation  of  kinetic  energy  and 
work: 

2hnv^  -  :^hnvo^  =  S  Hxdx  +  Ydy  -\-  Zdz).         (3) 

If  in  particular,  there  exists  a  force  function  or  potential  U 
for  the  forces  X,  Y,  Z,  the  system  is  said  to  be  conservative. 
We  then  have 

i:(Xdx  +  Ydy  +  Zdz)  =  dU, 

so  that  (3)  becomes  in  the  usual  notation  (V  =  —  U): 

T  +  V  ^  To+  Vo  =  const.; 

this  expresses  the  principle  of  the  conservation  of  energy. 


348  KINETICS  1470. 

470.  A  system  of  n  particles  possesses  a  centroid  whose 
co-ordinates  x,  y,  z  at  any  instant  are  given  by  the  equations 

Mx  =  Sma;,     My  =  Sm?/,     Mz  =  Sms;, 

where  M  =  'Em.  The  principles  of  the  conservation  of  linear 
and  angular  momentum  (Arts.  363,  366)  are  found  to  hold 
just  as  for  a  rigid  body. 

Thus,  in  the  case  of  the  solar  system,  if  the  action  of  the 
fixed  stars  be  neglected,  the  centroid  of  the  system  must 
move  uniformly  in  a  straight  line  and  there  exists  an  "  in- 
variable plane"  (Art.  367). 

2.  Constrained  system. 

471.  In  the  case  of  a  system  of  particles  subject  to  con- 
straints or  conditions,  we  may  try  to  replace  the  conditions 
by  constraining  forces  or  reactions  after  the  introduction  of 
which  the  system  can  be  treated  as  free.  The  equations 
of  motion  of  the  particle  m  will  then  again  have  the  form 
(1),  Art.  468;  ])ut  the  right-hand  members  now  contain  the 
unknown  reactions.  The  principle  of  virtual  work  gives 
d'Alembert's  equation  (2),  Art.  469;  and  the  virtual  dis- 
placement can  often  be  selected  so  that  the  unknown  con- 
straining forces  will  do  no  work  and  hence  will  not  appear 
in  equation  (2).  This  constitutes  the  main  advantage  of 
d'Alembert's  principle. 

472.  Before  proceeding  it  may  be  well  to  indicate  here  the  con- 
siderations by  which  d'Alembert  himself  (and,  in  more  exact  language, 
Poisson)  explained  his  celebrated  principle. 

Any  particle  m  of  the  system  is  acted  upon  at  any  time  t  by  two  kinds 
of  forces,  the  given  external  and  internal  forces,  whose  resultant  we 
denote  by  F  (Fig.  95),  and  the  internal  reactions  and  constraining 
forces  whose  resultant  we  call  F'.  The  resultant  of  F  and  F'  must  be 
geometrically  equal  to  the  effective  force  mj,  where  j  is  the  acceleration 
of  the  particle  at  the  time  t. 


473.]  MOTION  OF  A  SYSTEM  OF  PARTICLES  349 

Now,  if  we  introduce  at  m  the  equal  and  opposite  forces  mj,  —  mj, 
the  motion  of  the  particle  is  not  affected.  But  we  can  now  replace  F 
and  —  mj  by  their  resultant  F";  and  as  F,  F',  —  mj  are  in  equilibrium, 
so  are  the  forces  F'  and  F";  i.  e.  F"  is  equal  and  opposite  to  F'. 


Fig.  95. 

The  figure  shows  that  F  can  be  resolved  into  the  components  mj 
and  F";  the  former  produces  the  actual  change  of  motion  of  the  particle 
while  the  latter  is  consumed  in  overcoming  the  internal  reactions  and 
constraints  represented  by  F'.  This  component  F"  of  F  is  therefore 
called  by  d'Alembert  the  lost  force.  As  F'  +  F"  =  0  at  every  particle 
of  the  system,  d'Alembert's  principle  can  be  expressed  by  saying  that, 
at  every  moment  during  the  motion,  the  lost  forces  are  in  equilibrium 
with  the  constraints  of  the  srjstem. 

If  the  constraints,  instead  of  being  expressed  by  means  of  forces,  are 
given  by  equations  of  condition  we  may  express  the  same  idea  by 
saying  that,  owing  to  the  given  conditions,  the  lost  forces  forin  a  system  in 
equilibrium. 

473.  We  shall  now  assume  that  the  constraints  or  condi- 
tions to  which  the  system  is  subject  are  expressed  by  means 
of  equations  (the  case  of  conditions  expressed  by  inequalities 
is  excluded)  between  the  co-ordinates  x,  y,  z  of  the  particles 
and  the  time  t;  such  systems  are  called  holono7nic.  If  the 
equations  contained  the  velocities,  the  system  would  be 
called  non-holonomic. 

A  simple  illustration  of  the  difference  between  the  two  is 
furnished  by  a  sphere  moving  on  a  plane.     The  position  of 


350  KINETICS  [474. 

the  sphere  can  be  determined  by  the  co-ordinates  x,  y  of  its 
center  and  Euler's  angles  d,  if,  ^p  (Art.  434).  If  the  plane  is 
smooth  the  system  is  holonomic ;  if  it  is  so  rough  as  to  prevent 
slipping,  X,  y,  Q,  <p,  ^  are  no  longer  independent,  and  the 
system  is  therefore  non-holonomic. 

474.  Let  there  l^e  k  conditions 

<p{t,xuyi,zi,x->,---)=^,    Kt,xi,yi,zi,xi,- ■■)=(),    •••    (4) 

for  a  holonomic  system  of  n  particles.  The  number  of  inde- 
pendent equations  of  motion  will  be  3n  —  A;. 

For,  these  equations  must  express  the  equilibrium  of  the 
given  forces,  together  with  the  reversed  effective  forces,  under 
the  given  conditions;  and  for  this  equilibrium  it  is  sufficient 
that  the  virtual  work  should  vanish  for  any  displacement  com- 
patible with  the  conditions,  the  work  of  the  reactions  and  con- 
straining forces  being  zero  for  such  virtual  displacement.  In 
other  words,  in  d'Alembert's  equation  (2),  Art.  469,  the 
constraining  forces  clue  to  the  conditions  will  not  appear  if 
the  displacements  5x,  by,  bz  be  so  selected  as  to  be  compati- 
ble with  the  k  conditions  (4) .  Now  this  will  be  the  case  if 
these  displacements  are  made  to  satisfy  the  equations  that 
result  from  differentiating  the  conditions  (4),  viz. 

l^{iP::bx+ipyby-\-<pzbz)=Q,     ^{ypM-{-^yby+ypzbz)=0,     •••     (5) 

As  in  Art.  347,  t  is  regarded  as  constant  in  this  differentiation. 
Indeed,  when  the  conditions  contain  the  time,  a  virtual  dis- 
placement is  defined  as  one  satisfying  the  conditions  (5). 

475.  By  means  of  the  k  equations  (5),  k  of  the  3n  dis- 
placements bx,  by,  bz  can  be  eliminated  from  d'Alembert's 
equation  (2).  The  remaining  3n  —  k  =  m  displacements 
are  arbitrary;  their  coefficients  must  therefore  vanish  sepa- 
rately; equating  them    to  zero  we    have    the  2>n  —  k  =  m 


476.]  MOTION  OF  A  SYSTEM  OF  PARTICLES  351 

equations  of  motion  of  a  system  of  n  particles  with  k  conditions. 
To  do  this  more  systematically  we  may,  as  in  Arts.  348, 
351,  use  Lagrange's  method  of  indeterminate  multipliers: 
adding  the  equations  (5),  multiplied  by  X,  /x,  •  •  •  ,  to  d'Alem- 
bert's  equation  (2),  we  obtain  a  single  equation  in  which  the 
k  multipliers  X,  n,  •  •  •  can  be  selected  so  as  to  make  the 
coefficients  of  k  of  the  Sn  displacements  8x,  By,  8z  vanish. 
The  remaining  3n  —  k  displacements  being  arbitrary  their 
coefficients  must  likeAvise  vanish.  Hence  the  coefficients  of 
all  the  displacements  must  be  equated  to  zero,  and  this  gives 
n  sets  of  3  equations  of  the  type 

7nx  =  X  +  X«^x  +  M'Ax  +  •  •  •  7 

mi)  =   F  +  \^y  +  fxxlyy  +  •  •  •  ,  (6) 

mz  =  Z    -]-  \ip^  +  fjL\p;  -\-  •  • '  . 

These,  together  with  the  equations  (4) ,  are  sufficient  to  deter- 
mine the  3n  co-ordinates  x,  y,  z  and  the  k  multipliers  X,  M;  •  •  •  • 
It  is  apparent  from  the  equations  (6)  that  the  constraining 
force  acting  on  the  particle  m  has  the  components: 

X'    =    \(p-c  +  ljL\pjc  +    •  •  •    , 

Z'   =  \<Pz  +  M'/'^  +   •  •  •  . 

476.  If  the  conditions  (4)  do  not  contain  the  time  the 
actual  displacements  dx,  dy,  dz  of  the  particles  can  be  taken 
as  virtual  displacements;  and  d'Alembert's  equation  then 
gives  the  equation  of  kinetic  energy  and  work 

dlhnv^  =  ^(Xdx  -f  Ydy  -\-  Zdz).  (7) 

This  also  follows  from  the  equations  (6)  after  multiplying 
them  by  xdt,  ydt,  zdt  and  adding.  For,  the  coeffici(>nts  of  X, 
/i,  •  •  •  in  the  resulting  equation,  viz.  '^X^i-x  +  ipyij  -f-  (pzz)dt, 
'Li^iX  -\-  \pyy  +  \pzz)dt,  •  •  •    are  zero  as   appears    by  differ- 


352  KINETICS  [477. 

entiating  the  conditions  (4)  with  respect  to  t.  This  means 
that  the  constraining  forces  in  this  case  do  no  work  in  the 
actual  displacement  of  the  system,  as  they  are  all  perpen- 
dicular to  the  paths  of  the  particles. 

If,  however,  the  conditions  (4)  contain  the  time  explicitly, 
their  differentiation  gives 

so  that  instead  of  (7)  we  find: 

d^imv^  =  S(Xda;  +  Ydy  +  Zdz)  -  \<ptdt  -  ixxPtdt-  •  •  •  .   (7') 

477.  If  the  conditions  (4)  do  not  contain  the  time  and  if, 
moreover,  a  force-function  U  =  —  V  exists  for  all  the  forces 
we  find  as  in  Art.  469  the  principle  of  the  conservation  of  energy: 

T+V  =  To+  Vo. 
But,  even  if  no  force-function  exists,  the  elementary  work 
'Z{Xdx  -{-  Ydy  -{-  Zdz)  is  a  quantity  independent  of  the  co- 
ordinate system,  and  J  '^{Xdx  -f-  Ydy  +  Zdz)  =  W,  say, 
represents  the  work  done  by  the  external  and  internal  forces 
in  the  time  t;  we  have  therefore: 

^hnv^  -  Zhnvo'^  =  W. 

3.  Generalized  co-ordinates;  Lagi'ange's  equations 
of  motion;  Hamilton's  principle. 

478.  As  shown  in  Art.  358,  the  number  of  conditions  that 
make  a  system  of  n  particles  invariable,  i.  e.  make  it  a  free 
rigid  body,  is  fc  =  3n  —  6.  A  free  rigid  body  has  therefore 
3n  —  fc  =  6  independent  equations  of  motion. 

A  rigid  body  with  a  fixed  axis  (Art.  415)  has  but  1  degree 
of  freedom  and  5  constraints;  its  motion  is  given  by  a  single 
equation. 


479.]  MOTION  OF  A  SYSTEM  OF  PARTICLES  353 

A  rigid  body  that  can  turn  about  and  also  slide  along  a 
fixed  axis  (Art.  235)  has  4  constraints  and  2  degrees  of 
freedom;  it  has  2  equations  of  motion,  its  position  being 
determined  by  2  co-ordinates,  say  the  angle  6  and  the  distance 
X  measured  along  the  axis. 

A  rigid  body  with  one  fixed  point  (Art.  233)  is  an  example 
of  an  invariable  system  with  3  constraints  and  3  degrees  of 
freedom.  Three  variables  (such  as  Euler's  angles  6,  <p,  i/', 
Art.  434)  are  necessary  and  sufficient  to  determine  a  particular 
position,  and  the  number  of  independent  equations  of  motion 
is  3. 

479.  These  considerations  can  be  generalized  so  as  to  apply 
to  a  general  variable  system  of  n  particles  with  k  holonomic 
conditions.  Such  a  system  is  said  to  have  on  —  k  ^  m 
co-ordinates  because  it  has  3n  —  /b  =  m  independent  equa- 
tions of  motion  (Art.  474).  In  other  words,  in  the  place 
of  the  3/1  cartesian  co-ordinates  x,  y,  z  of  the  n  particles,  sub- 
ject to  k  conditional  equations  (4),  we  may  introduce 
Zn  —  k  —  m  independent  variables,  say  gi,  •  •  •  q^,  which 
are  selected  so  as  to  satisfy  the  k  conditions  (4)  identically. 
These  vairables  are  called  the  generalized,  or  lagrangian, 
co-ordinates  of  the  system  (comp.  Art.  352). 

Suppose,  for  instance,  that  the  system  is  subject  to  only 
one  condition,  viz.  that  the  point  Pi  of  the  system  should 
remain  on  the  surface  of  the  ellipsoid 

_  a;i2      7/i2      2i2 

^    =    ^2"      I      T2"  +  '    %     —1=0, 

o/-        b^        c^ 

If  we  select  two  new  variables  qu  q^,  connected  with  Xi,  y\,  Zi 
by  the  equations 

Xi  =  a  cosgi,     yi  =  h  smqi  cosqz,     Zi  =  c  singi  sinqz, 
24 


354  KINETICS 


1480. 


the  condition  v?  =  0  is  satisfied  identically  in  the  new  co- 
ordinates qi,  52-  Hence,  by  introducing  qi,  q^  in  the  place 
of  xi,  iji,  Z\  the  condition  ^  =  0  is  eliminated  from  the 
problem. 

The  motion  of  a  system  with  m  degrees  of  freedom  in 
ordinary  three-dimensional  space  might  be  interpreted  as 
the  motion  of  a  free  particle  in  a  space  of  m  dimensions. 

480.  The  introduction  of  the  generalized  co-ordinates  q\, 
•  •  •  gm  of  a  system  with  m  =  Sn  —  fc  degrees  of  freedom  into 
the  equations  of  motion  (G),  Art.  475,  is  performed  just  as 
the  corresponding  prol^lem  in  Arts.  353-355. 

The  cartesian  co-ordinates  x,  y,  z  of  any  one  of  the  n  par- 
ticles are  given  functions  of  gi,  •  •  •  q„i  and  of  t  so  that 

.  _  dx   .     .  .    dx   .        dx 

dqi  ^  dqm  dt 

with  similar  expressions  for  y,  z.  Hence,  on  the  one  hand  we 
have  if  q  denote  any  one  of  the  co-ordinates  qi,  ■  ■  •  qmi 


(8) 


dx       dx 

dy      dy      dz      dz  ^ 

dq  ~  dq' 

l)q       dq'     dq      dq 

on  the  other: 

dx        d'^x    .     , 
^— -  ^  ^     ^1  +  ••• 
dq      dqdqi 

d"X     .           d^X 
'^  dqdqm^'"  ^ dqdt 

=  ^'dq,dq+-" 
that  is: 

d    dx       d  dx  _  d  dx 
'^  ^"^  dqmdq '^  dtdq  ~  dtdq' 

dx      d  dx 

dy  _  d  d]f      dz       d  52^ 

dq  ~  dtdq' 

dq       dtdq'     dq       dtdq' 

(9) 


With  the  aid  of  the  relations  (8)  and  (9)  we  find  for  the 
derivatives  of  the  kinetic  energy  T  =  2i-m(x^  +  y-  +  z~) : 


481.1  MOTION  OF  A  SYSTEM  OF  PARTICLES  355 

dq   -^"^Vdq^^dq    ^  '  dq  ) 

^     /  .  d  dx  ,    .  d  dif   ,    .  d  dz\ 

ST     ^    I M  ,  .ay  ,  .az\     „    I  .ax  ,  .ay  ,  .az\ 


hence 


ddT      ^     /  ..dx   ,..dy   ...dz\    .dT  .,„. 

dt-e^  =  '''^V'dq+ydq  +  '6q)  +  eq-  ^^^^ 


Now  multiplying  the  equations  of  motion  (6)  by  dx/dq, 
dy/dq,  dz/dq  and  adding  them  throughout  the  whole  system 
we  find 

the  coefficients  of  X,  ju,  •  •  •  being  all  zero  since,  by  hypothesis, 
the  new  co-ordinates  satisfy  the  conditions  (4)  identically. 
The  right-hand  member  of  (11)  will  be  denoted  by  Q 
(comp.  Art.  353) ;  substituting  for  the  loft-hand  member  its 
value  from  (10)  we  find  Lagrange's  equations  of  motion: 

ddT_dT_ 

dtdq        dq   ~^'  ^^^^ 

As  there  is  one  such  equation  for  each  of  the  lagrangian  co- 
ordinates qi,  •  •  •  qm,  their  number  is  wi  =  3n  —  k.  They  are 
obtained  from  the  type  (12)  by  attaching  successively  the 
subscripts  1 ,  •  •  •  w  to  g,  ^,  and  Q. 

481.  In  the  particular  case  of  a  conservative  system,  i.  e. 
when  there  exists  a  force-function  U  such  that 

2x=t-,  ^y-f,  sz=f, 

dx  dy  dz 


356  KINETICS 


1482. 


we  have  Q  —  dU/dq;  the  equations  of  motion  then  have  the 
form 

This  relation  can  be  derived  directly  from  the  equation  (10) 
by  ol:)serving  that  in  any  infinitesimal  displacement  the  work 
of  the  effective  forces  is  equal  to  the  increase  of  U  (or  decrease 
of  the  potential  energy  V  =  —  U).  Now  if  we  vary  the 
co-ordinate  q  alone  by  8q,  the  variations  of  x,  y,  z  are  {dxldq)bq, 
(dyldq)8q,  {dz/dq)8q;  hence  the  work  of  the  effective  forces 
7nx,  my,  mz  is 

the  first  term  in  the  right-hand  member  of  (10)  is  therefore 
=  dU/dq,  and  this  at  once  gives  (12'). 

482.  Finally,  if  we  denote  the  function  T  -\-  U,  that  is,  the 
difference  T  —  F  of  kinetic  and  potential  energy,  by  L: 

L=T+U=T-V, 

and  observe  that  U  is  independent  of  the  velocities  so  that 

dL^dT 
dq  ~  dq' 

the  equations  of  motion  can  be  written  in  the  simple  form 

dtdq       dq'  ^  "  ^ 

in  which  they  depend  on  a  single  function.     This  function  L 
is  called  the  kinetic  potential  (according  to  Helmholtz)  or 

the  lagrangian  function. 


485.]  MOTION  OF  A  SYSTEM  OF  PAHTICLES  357 

483.  To  illustrate  the  use  of  Lagrange's  equations  let  us  derive  the 
equations  of  motion  in  polar  co-ordinates. 

In  the  case  of  plane  motion  we  have 

T  =  lm{f-^  +  r292), 
whence 

dT         .      d  dT         ..      dT 

dr  at  dr  dr 

The  left-hand  member  of  (12)  is  therefore  m{f  —  rO^).      The  right-hand 

member  Q  is  determined  by  observing  that  Qbq  is  the  work  of  the 

forces  in  the  displacement  bq.     Hence  in  our  case,  if  R  is  the  resultant 

force,  Rr  and  Re  its  components  along  and  at  right  angles  to  the  radius 

vector  r  (Art.  269),  Qbr  =  Rrbr,  i.  e.Q  =  Rr.     Hence  the  first  equation 

of  motion  is 

7n{r  —  rg-)  =  R,-. 

To  find  the  second  equation  we  have 

BT  „■      dT      ^ 

as  Qbd  is  the  work  done  on  the  particle  when  B  varies  by  hQ,  i.  e.  in  the 
displacement  rhd  at  right  angles  to  the  radius  vector,  we  have  Qbd  = 
RgrSd;  hence  the  second  equation 

mjir'^b)  =  rRe. 

484.  For  polar  co-ordinates  r,  0,  </>  in  three  dimensions  (Art.  269)  we 
have 

T  =  i //?.('■•-  +  r-b"-  +  r2sin26>  -p^), 
and  we  find: 

vi\f  —  r(d-  +  S\D?d  <p')\   =  Rr, 

m    -r  (r'^'d)  —  r-  sm^  cosi9  <p-      =  rRe, 

m  J  (r^  sin^^  <p)  =r  smO  R^. 

If  there  exists  a  force-function  U  the  right-hand  members  are  dU/Br, 
dU/dd,  dU/D^. 

485.  As  another  example  consider  the  motion,  in  the  vertical  xtj- 
plano  of  two  particles  P,  P'  (Fig.  96)  of  masses  m,  m',  suspended  by 


358 


KINETICS 


[485. 


0 

X 

1 

/ 

/' 

/ 

y 

/ 

y 

/ 

a  1 

1  / 

'0 

J/' 

y 

/? 

< 

y 

Fig.  96. 

weightless  rods  OP,  PP'  of  lengths  a,  b.  If  the  co-ordinates  of  P,  P' 
are  x,  y  and  x' ,  y'  and  the  inclinations  of  OP,  PP'  to  the  vertical  d,  <p, 
we  have 

X  =  a  sin0,     y  =  a  cosO, 

x'  =  a  sine  +  b  sm<p,     y'  =  a  cosO  +  b  cosip, 
whence 

U  =  mgy  +  m'gy'  =  g[{»i  +  ?n')a  cose  +  tn'b  cos<p]. 

Denoting  by  v,  v'  the  velocities  of  P,  P'  we  have 

ifl  =  a?&^,     ?/2  =  a-e-  +  ¥^^  +  2ab  cos(<p  -  6)6 ip, 
whence 

T  =  \[{m  +  m')a-e'^  +  m'b'^^-  +  2m'ab  cos(^  -  0)0^]. 

Lagrange's  equations  are  then  found  to  be 
d 
dt 


{{m  +  m')a^O  +  m'ab  cos{<p  —  6)  if] 


d 


—  m'ab  sin(v?  —  9)6ip  +  g{m  +  m')a  sin©  =  0, 
j?n'[b^,p  +  ab  cos{<p  —  fl)e]  +  7)i'ab  sm{<p  —  e)6ip  +  gm'b  sin^=  0. 

If,  in  particular,  6,  <p,  6,  Ip  are  so  small  that  their  third  powers  can 
be  neglected  the  equations  reduce  to 

(m  +  m')(a  +  m'b'^  +  (wj  +  m!)g6  =  0, 

oB  -\rb'i(>  Ar  gf  =  ^^ 
To  mtegrate  put  i9  =  A  cosr/,  ip  =  \A  cosrt,  whence 

(m  +  7n')(g  —  ar-)  =  m'bXr"^,     \(g  —  br~)  =  ar-, 
mabr^  —  {m  +  ■m')g{a  +  b^-  +  (?«  +  vi')g-  =  0. 


486.]  MOTION  OF  A  SYSTEM  OF  PARTICLES  359 

The  last  equation  regarded  as  a  quadratic  in  r-  has  real  (the  discriminant 
being  positive)  and  positive  roots,  say  r-  and  rr.  Corresponding  to 
these  roots  we  have 

e  =  A  cosrt,     (p  = .„  cosrt, 

g    —    f)j-2  ' 

e  =  Ax  cosnt,    <p  = — -  cosri<. 

g  —  oTi^ 

In  the  same  way,  by  substituting 

e  =  A  sinrt,     ^  =  X.4  sinr^ 

we  obtain  the  special  solutions 

A'ar'' 

e  =  A'  sinrt,     <p  = ,  „  sinrt, 

g  —  br^ 

A  ,   ■      ,  Ai'ar\~    . 

6  =  Ai  snird,    <p  = ,    „  smri^ 

g  -  brr 

The  general  solution  is  of  course  the  sum  of  the  four  special  solutions, 
and  the  four  constants  are  determined  from  the  initial  positions  and 
velocities  of  m  and  m'. 

486.  From  Lagrange's  equations  (12),  Art.  480,  it  is  easy 
to  derive  Hamilton's  principle. 

Let  each  of  the  m  equations  (12)  be  multiphed  by  the 
infinitesimal  displacement,  or  variation,  8q;  let  the  equations 
be  added,  multiplied  l3y  dt,  and  integrated  from  ^i  to  to: 

The  first  term  can  be  transformed  by  integration  by  parts; 
as  d(8q)/dt  =  8(dq)/dt  we  have 

If  now  the  variations  Sq  be  selected  so  as  to  vanish  ])oth  at  the 
time  ^1  and  at  the  time  ^o,  the  first  tern^  on  the  right  vanishes 
at  both  limits.     Hence  the  equation  (13)  assumes  the  form 


360                                         KINETICS  [487. 

As  2(^  —  65  +  ^^55)  =57^  and  ZQ8q  =  8U  for  a  con- 
servative system  and  =  8W  (the  elementary  work)  for  any 
system,  the  equation  reduces  to  the  simple  form 

(Sr  +  8W)dt  =  0  (14) 


in  the  general  case,  and 

5  f\T-\-U)dt  =  0,or8  f  \T-V)dt  =  0,  or  8  ]Ldt  =  Q\,    (14') 
Jti  tJti  fJti 

for  a  conservative  system. 

487.  Hamilton's  principle  consists  in  the  proposition  that 

the  equation  (14)  or  (14')  holds  for  any  virtual  displacements 

of  the  system  that  are  zero  at  the  times  ti  and  h. 

'Ldt  =  0, 

the  principle  is  often  expressed  briefly  by  saying  that  for  the 

actual  motion  the  mean  value  of  the  kinetic  j)otential  L  —  T  —  V 

in  any  time  to  —  ti  is  a  rninimmn  as  compared  ivith  other 

motions  between  the  same  tivo  configurations.     More  exactly 

we  can  only  say  that  the  variation  of  this  mean  value,  i,  e. 

of  the  integral  J^  'Ldt,  is  zero. 

A  more  complete  discussion  of  Hamilton's  principle  and 
of  the  somewhat  similar  principle  of  least  (or  stationary) 
action  will  be  found  in  E.  T.  Whittaker,  Analytical  dynam- 
ics, Cambridge,  University  Press,  1904. 


ANSWERS. 

Art.  7. 

(1)   (a)  5.87;  (b)  40.62;  (c)  58.67;  (d)  25.38;  (e)  1086.9; 


(/)  82,020;  (g)  9.84  X  lO^. 

(2)  t  =  2{a  +  h)l{v,  +  V2). 

(3)  184,200  M./sec. 

(4)  35  M./h. 


(5)  i 

(6)  18f. 

(7)  10.4. 


Art.  9. 

(1)  (a)  a;  {h)  2at+h;  {c)iaf^;  id)-ak  smkt;  (e)-ae-'; 
(/)  ia(e'-0;  (g)  ai(^  +  l);  W  4a^(i2_i).  (^)  af(3f-2); 
(j)  5af(i3_8).  (^)  a(l  +  ^^)/(l  -  tr~- 

(2)  (a)_^o_+ vo^  +  ig^^;  (&)  So-  4:at  +  ^aP;  (c)so  +  atmt; 

(d)  SoVl  -  r-;  (e)  So  +  (a/a)e3(e«'  -  1). 

(3)  (a)  s  =  So  +  vot  -f-  i{/^-;  (6)  s  =  a  sint;  (c)  s  = 
ia(e'  —  e~0. 

Art.  12. 

(1)  0.73.  (3)  0.11  ft./sec.2. 

(2)  32.185.  (4)  Jig  =  1/293. 

(5)  (a)  0;   (h)  2a;  (c)   -  a/4t^;   {d)   -  ak^  coskt  =  -  k^s', 

(e)  ae-'  =  s;  (/)  ia(e'  +  e"')  =  s;  (gf)  a{2t  +  1); 
(/i)  4a(3f2  _  1).  (^)  2a(3«  -  1);  (j)  20a(i3  -  2);  ik) 
2at(P  +  3) /(I  —  Py. 

(6)  (a)  g;  (6)  2at;  (c)  2a  sin^/cos^^;  (d)  -  so(l  -  ^'H; 
(e)  aae'"'"*"^. 

Art.  19. 


(1)  (a)  128.8;  (b)  257.6;  (e)  144.9. 

(2)  0.0917. 


(4)  h  =  c 


6f 


-^/K^ 


2f+'^ 
(I 


An    approximate 


value  \s  h  =  ^gcf^Hc  +  gO-  For  a  direct  numerical  compu- 
tation the  method  of  successive  approximatiojis  can  be  used: 
neglecting  ti  find  /i  approximately  from  h  =  i^//^  with  ^  =  4; 

361 


362  ANSWERS 

with  this  value  of  h  find  1-2,  hence  the  time  ti,  with  which 
correct  A;  etc.     Result:  h  —  70.4  m. 

(5)  (a)  4f  min.;  (6)  0.18;  (c)  49^;  (d)  4  min.  4^  sec. 

(6)  (a)  I'o/s';  (&)  ii'oVS';  (c)  2t'o/6';  (^)  -  Wo. 

(7)  (a)  4J  M.;    (6)  645  ft./sec;    (c)   li  min.;    (d)   1200 
ft./sec;  (e)  17,  58  sec.  (8)  80  ft./sec. 

(9)  (a)  h/vo;  (6)  /i  -  ighVvo';  (c)    a^. 

(10)  338,000  ft./sec;  jh  sc.       (12)  30  M./h. 

(11)  426  ft.  (13)  I  ft. 

Art.  24. 

(1)  5M./sec.;  34i  min. 

(3)   (a)  7  M./sec;  (b)  7  M./sec,  4  days  20i-  hours. 


(4)  .  =  -  V2gi^^f  ^i,.  where  ^^-|-  -|- 

(a)  If  yo<  ^l2^Jf,  t=---iL=.\  Mkm-s)-  ^so{km-So) 

+  ^-7?(cos-^^-cos-^^|)]; 

(6)  if.o=V2^Jf  ,  ^  =  iJ^4(so^-5^); 
\  So  ^  9  ^ 

(c),f».>V2,«^-,(=^=[- 


fc2/^  log 


V2yi2  L  Vso  +  ^^2i^  +  Vso 


+  Vso(so  +  k'"R)  -  Ms  +  km) 

(5)  If  ?^o  <  -^2gR  the  height  above  the  earth's  surface  to 
which  the  particle  rises  is  /i  =  Vo^RI(2gR  —  vo^)  and  the  time 
of  rising  to  this  height  is 

R         (       .  2gR  .  _,     vo 


2gR  -  Vo~  \  ^i2gR  -  v}  ^l2gR  /  ' 

if  Vo  =  ^2gR,  the  time  of  rising  to  the  distance  s  from,  the 
center  is 


ANSWERS  363 


SR^i2g 


and  the  particle  does  not  return;  if  vo  >  ^2gR  the  time  is 
-S^gR  L  Vs  +  Vs  +  fc^    J 


where  /c''  — 


Wo'  -  2gR ' 
(6)  /t  =  72,  ^  =  (1  +  ^tt)  V^  =  34f  min. 

Art.  28. 

(2)  V  =  ^JgR  =  5  M./seC;  T  =  1  h.  25  min. 

(3)  Vso^  +  (volixr. 

Art.  36. 

(1)  tt;  15.7  ft./sec.  (3)   (a)  402;  (b)  25.1  sec. 

(2)  0.157  rad./sec.2;  5  rev. 

(4)  (a)  0.022  rad. /sec;  (6)  15.7  ft./sec;  7.85  ft./sec. 

Art.  42. 

(l)-(5)  Check  graphically.        (7)  36M./li.;  198  ft. 

(6)  20".  (8)  Vr  =  Vb  sine. 
(9)  Spiral  of  Archimedes  r  =  (vo/w)  d. 

Art.  48. 

(4)  The  projection  of  the  velocity  on  the  radius  vector  and 
on  the  focal  axis  are  in  the  constant  ratio  e  of  the  focal  radius 
to  the  distance  to  the  directrix.  It  follows  that  the  tangent 
meets  the  directrix  at  the  same  point  as  does  the  perpen- 
dicular to  the  radius  vector  through  the  focus. 

Art.  56. 

(7)  j2  =  a"[2(l  -  COS0)  ^'  +  2  sin0  er-  +  6']. 

(8)  (a)  Straight  line;  (6)  circle;  (c)  circle  of  radius  v;  (d) 
it  is  normal. 

(9)  Tlie  cylindrical  components  are 

j\  =  f'  -  rV",     h  =  2r'<p  +  r'^,    jz  =  x, 


364  ANSWERS 

Avhere  r'  =  r  cos0.     The  spherical  components  are  found  by 
projection: 

>  =  ji  sine  +  J3  cos^,    je  =  ji  cos^  -  Ja  sin0,    j^  =  > 

Art.  59. 

(3)  45°. 

(5)  Construct  a  vertical  circle  having  the  given  point  as 
its  highest  point  and  touching  (a)  the  straight  line,  (6)  the 
circle. 

Art.  61. 

(9)  (a)  1374-  ft.  from  the  vertical  of  the  starting  point; 
(6)  6i  sec;  (c)  201  ft./sec,  at  6^°  to  the  vertical. 

(10)  227.5  ft./sec.  (11)  4°  22'  or  86°  48'. 

(13)  Let  Oy  =  vo  be  the  given  initial  velocity.  On  the 
vertical  through  0  lay  off  0T>  =  H  =  Vo'^!2g;  then  the  hori- 
zontal through  jDis  the  chrectrix.  Make  2^  VOF  =  2^  DOF, 
and  lay  ofiOF  =  OD  =  H;  then  F  is  the  focus. 

(14)"  With  VoV2g  =  H,  the  locus  is  a;^  =  -  4:H(y  -  H), 
a  parabola.  (17)  A  horizontal  line. 

(18)  (a)  1.5  sec;  (h)  25.1  ft.  from  the  building;  (c)  59.7 
ft./sec,  at  16i°  to  the  vertical. 

(19)  300  ft.  from  tee,  in  1  see. 

(20)  At  a  distance  of  6250  ft. 

Art.  68. 

(1)  0.99672;  86117.  (4)  28.8  ft. 

(2)  3.26  ft.  (5)  980.4. 

(3)  32.158. 

(8)  The  pendulum  should  be  lengthened  by  y^^  of  its 
length. 

(9)  It  will  lose  67  sec. /day.      (10)  About  a  mile. 

Art.  70. 

(3)  1.0038. 

(5)  Use  the  first  formula  of  Art.  69. 

(6)  Determining  the  constant  from  6  =  r  for  v  =  0  we 
have  iw^  =  2gl  cos^i^.  Putting  v  —  —  Idd/dt  and  inte- 
grating  gives  t  =  -yjl/g  log  tanj(7r  -\-  6)  if  0  =  0  for  i  =  0. 


ANSWERS  365 

Hence  the  bob  approaches  the  highest  point  of  the  circle 
asymptotically,  i.  e.  without  reaching  it  in  any  finite  time. 

Art.  75. 
(1)  X  =  Xq  cosfxt  +  (volfj.)  smfit. 


{2)  V  =   -  |x^la^  -  x\ 


Art.  80. 


(1)  X  =  10.806  cosCiTT^  +  271°). 

(2)  X  =  2a  cosiS  cos(cof  +  i5). 

(3)  (a)  X  =  2acoscof;  (6)  a;  =  0,  the  case  known  in 
physics  as  interference. 

(4)  xi=  -  5.18  coswf,  xo  =  14.14  cosM  +  30°). 

Art.  113. 

(1)  a;^  +  t/^  =  a-  being  the  circle,  j  =  —  ahi^jif  where  Vi 
is  the  x-component  of  the  initial  velocity. 

(2)  WoVa. 

(3)  Let  j  =  )uV;  then,  if  (xo,  v/o)  is  the  initial  position,  I'l, 
??2  the  components  of  the  initial  velocity,  the  path  is  the 
hyperbola : 


(i'2--/i-?/o-).c-+2(/x2xo2/o-i'i?'2)  +  0'r-AtW)|/^=(y2a;o-?^iyo)^ 

,-.  iJ.         r.      Voro  sim/'o         ,  Vo^  sin2;/'o 

(5)  a  =  —  ,       6  = ,       tana  = ^- ^ .rj^  , 

^  t^  e  t^  -\-  Vo^  cos2i^o 

where  e^  = v^"^ . 

To 

(6)  Put  r  =  1/m  and  determine  d^u/dd"^  in  terms  of  u  alone: 

fj~')i 

~=  -  u-\-  (n  -  1)(1  -  e'~)q-^"u-^'"+'  -  (n  -  2)g-"w-"+i. 

Hence  by  (IG),  Art.  100: 

/(r)  =  c-iin  -  1)(1  -  e2)5-2"ir2"+'  -  (/i  -  2)q-"u-"+^. 

n  =  1  gives  an  ellipse  if  e  <  1,  a  parabola  if  e  =  1,  a  hyper- 
bola if  e  >  1,  all  referred  to  focus  and  focal  axis;  n  =  2  gives 
conies  referred  to  their  axes;  n  =  —  1  gives  pascalian  lima- 


366  ANSWERS 

gons  (cardioids  for  e  =  ±1);  n  =  — 2  gives  a  lemniscate 
if  c  -  ±  1. 

(7)  (a)  c2(2aV-5  +  r-3);    (6)  cV-^;    (c)    c-(l  +  n2)r-3;    {d) 

c-[2?i"aV~^  +  (1  —  n^)r"^]. 

(8)  Sa^cV-^. 

(9)  Ellipse,  parabola,  or  hyperbola  according  as  ju  ^ 
2'2^I/o^,  2/0  being  the  initial  distance  from  the  plane,  V2  the 
component  of  the  initial  velocity  normal  to  the  plane. 

(10)/(r)  =  --,^. 

a^    y3 

Art.  137. 

(3)  The  direction  of  motion  passes  through  the  highest 
point  of  the  wheel. 

(4)  With  the  center  0  of  the  given  circle  as  origin  and  the 
perpendicular  to  I  through  0  as  axis  of  x,  the  fixed  centrode 
is  y^  =  ex  =^  a  -yjx^  +  ?/'^  where  a  is  the  radius  of  the  given 
circle,  c  the  distance  of  0  from  I.  With  A  as  origin  and  h  as 
axis  of  X  the  body  centrode  h  x^  =  ay  ^  c  ^x'^  +  y"^-  The 
upper  sign  corresponds  to  h  sliding  over  the  first  and  second 
cjuadrants  of  the  circle,  the  lower  to  h  sliding  over  the  third 
and  fourth  quadrants.  If  c  >  a,  the  complete  fixed  centrode 
has  a  node  at  0  with  the  tangents  ay  =  ±  Vc^  —  a'^x.  The 
polar  equations  of  the  centrodes  are  r  sin-0  =  c  cos0  +  a  and 
r'  cos'^9'  =  a  sin0'  +  c.  The  body  centrode  for  c  >  a  is 
(apart  from  position)  the  same  curve  as  the  fixed  centrode 
for  a  >  c,  and  vice  versa. 

(5)  •//-  =  2a{x  +  ia). 

(6)  The  fixed  centrode  is  a  circle  passing  through  Oi,  O2; 
the  body  centrode  is  a  circle  of  twice  the  radius  of  the  fixed 
centrode.  The  path  of  any  point  in  the  fixed  plane  is  a 
Pascal  limagon;  the  points  of  the  body  centrode  describe 
cardioids. 

(8)  Two  equal  parabolas;  the  motion  is  the  same  as  that 
of  Ex.  (5). 

(10)  With  0  as  pole  and  OB  as  polar  axis  the  equation  of 
the  fixed  centrode  is  r-  cos-^  —  2ar  cos-9  +  a^  =  P.  With  0 
as   origin  and   OB   as   axis  of  x  the  equation  is   (x^  -j-  a^ 


ANSWERS  367 

—  P)  ^lx'^  +  Z/^  =  2ax'^.  The  rationalized  equation  represents 
the  centrode  of  A  5  when  B  moves  not  only  on  the  positive  but 
also  on  the  negative  half  of  the  axis  of  x.  The  equation  of 
the  body  centrode,  with  A  as  pole  and  A  5  as  polar  axis, 
AC  ^  r',  ^  BAG  =  6',  is  found  by  observing  that  r  = 
r'  -j-  a,  I  sine'  =  OB  sin0  =  r  cos6  smd  whence 

(a2  -  P  cos^e')^-''  -  2a/V  sin^e'  +  l-iP  -  a^  cos0')  =  0, 

i.  e. 

,      A  -\-  a  cos^'         ,      A  —  a  cos0' 

ri   =  I  — —. T7 ,     r2   =1 -. -, . 

a  +  i  cosO  a  —  i  com 

These  relations  can  be  read  off  directly  from  the  figure  if 
perpendiculars  be  dropped  from  0  on  AB  and  from  B  on 
AC.  For  the  path  of  any  point  P  whose  body  co-ordinates, 
with  A  as  origin  and  A  5  as  axis  of  x' ,  are  x' ,  y' ,  we  have 

x  =  a  cos^+rc'  COS97+?/'  sin^,     y  =  a  smd  —  x'  sin^+?/'  cos^?, 

where  6  and  ^  are  connected  by  the  relation  ?/a  =  sin^/sinc^. 
For  the  path  of  the  midpoint  oi  AB  we  find  x  =  a  cos0  + 
•2"^  coS(p,  y  =  a  sin0  —  ^l  shiip,  whence  x  =  -^a^  —  4?/-  + 
|-  -sP-  —  4?/^  which  is  of  the  fourth  degree. 

To  find  the  velocity  of  B  when  that  of  A  is  given  oljserve 
that  as  the  distance  AB  \^  invariable  the  projections  of  the 
velocities  of  A  and  B  owAB  must  be  equal,  whence  va  cos^  = 
Va  sin(0  +  >p). 

(12)  Find  first  the  velocity  Vr  of  Po  relative  to  Pi  as  the 
resultant  of  -i^i  and  v^;  hence  w. 

Art.  148. 
(2)  A  Pascal  lima^on. 

(7)   (a)  co-x  —  CO?/  =  0;  (b)  6:x  —  ory  -]- J  =  0. 

Art.  166. 

(2)  Distance  from  midpoint  =  \^l,  21  being  the  distance 
of  5  from  23. 

(3)  About  1000  M.  below  the  earth's  surface. 

(5)  X  =  r  sina/a  =  rc/s,  where  c  is  the  chord,  s  the  arc; 
for  the  semi-circle  x  =  (2/7r)r. 


368  ANSWERS 

.  ^     31/2  -  log(l  +  1/2)  ^  ^  ^^^^g 
*  1/2  +  log(l  +  1/2) 

Ot/2  —  1 

y  =  ^    ^-      '^    i — -^  a  =  1.12907a. 

^  l/2  +  log(l  +  i/2) 

(7)  Tra,  |a.  (8)  |a,  fa. 

(9)  r  sin^/9_,  r(l  -  cose)/0,  ^fcr  sin^. 

(10)  2a(Q!  sino: -f  cosa  —  1)/q:^,  2a(sinQ:  —  a  cosa)/a^. 

(12)  AV«,  ?rV«. 

(13)  Distance  from  hypotenuse  =  0.11a. 

(15)   (a)  ix„  %y,;  (b)  ^t,  ^t;  (c)  ^  a  =  0.40531a,   ^6; 

(17)  frsina/«.  (18)  \a. 

(22)  If  a:i,  X2  are  the  distances  of  the  planes  from  the  center 

then  a;  -  ,(a:i  +  x,)  ^,  _  ^^^^,  ^  ^^^^  _^  ^^,^ 

(a)   |(2a  -  /i)V(3a  -  /i);  (6)  fa;  |a(l  +  cosa). 

(23)  f/i.  (25)  t\7/i. 

(24)  A2/1-  (26)  fa,  §b,  fc. 

(27)(a)  |a,|a;  (6)'--p^a  =  1.85374a,    |a;    (c)    ffa; 

, ,.  128  +  454  _  .  . .  -  .__,      n  +  3 

(^^        907r       «  ==  ^'^^^^^^  ^28)  2-(^-^)  a. 

Art.  170. 

(1)  300,000  F.P.S.  units.  (3)  32.000  F.P.S.  units. 

(2)  50ft./sec. 

Art.  179. 

(1)  6.4  X  10^  poundals  =  8.9  X  10^  dynes. 

(2)  4.5  pounds.  (3)  0.1406. 

Art.  196. 

(1)  9  =  120°.  (3)  28,  39°  16'. 

(4)  2F  cos22|°  =  1M8F, 


ANSWERS  369 

(7)  (a)  W sine,  W cos0;  (6)  W  tsmd,  W  seed;  (c)  W sin^  seca, 
W  cos(0  +  a)  seca. 

(8)  W  siii)8/sin(a!  +  fi),  W  sinQ:/sin(a  +  /3). 

(10)  P  =  iW,T  =  iW- 

(11)  P  =  2W  cosJ(«  +  Itt)  =  0.8945TF. 

(12)  P  =  W  sm{a  +  j8)sin|S  is  greatest  when  the  sail 
bisects  the  angle  between  the  wind  and  the  direction  of 
motion. 

(13)  W  sin/3/sin(Q:  +  /S),  W  sinQ;/sin(a:  +  /3). 

(14)  T  =  Wl/d,  r  =  W{c  -  I) Id,  where  d^  =  V  -  i(c  -  ly. 

(15)  13.4,  28.9,  50,  86.6,  186.6,  oo. 

(16)  848,  282;  1000,  600.         (17)  0.640T^. 
(18)   (a)   1.414Pr;  (6)  2W  cosi(i7r  ±  6). 

Art.  221. 


(2)  T  =  mW,   A  =  Vm'  -  m  +  1   W,  where  m  =  2c//. 

(3)  F  =  i(cot0  -  rll)W. 

(4)  tan^  =  (a  cota  -  h  coti3)/(a  +  fc). 


(b)  A,= ^^ Tf,  ^.=  (  1--^  lF,fi=-^TF. 


Art.  243. 

(1)   P  =  TT  sm(plcos{a  —  if). 

(„)  sin(^^ri  <  P  <  sin(9i^  0  3=|a2sin«); 

cos(p  TF  cosv?      '  Ty 

(c)  if  P  act  up  the  plane.  p  =  sm(g+  (p)  ^    if  Pact  down 

COS(p 

,       ,          „  ^  sin(v?  —  d)  „, 
the  plane,  P  <  — ^^ ^  W. 

eos(p 

(4)  226i,  56*-.  (5)  B  =  ^t  -  2<p. 

(6)  fjL  =  I  sine  cos0/(c  -  I  cos^^). 

(7)  A  =  mW  sin(0  —  ^)  cos^/sinv?,  C  =  mTT  cos0,  tan2^ 
=  m  s\n20,  where  m  =  l/c. 

(8)  sin^^f. 


25 


370  ANSWERS 

Art.  247. 
(1)  33  X  10-12. 

Art.  254. 


(8)   '^^— [  Vc2  +  (a  -{-W  -   ^lc^+~(a:^^^l 


r      p  -\-  c  p 

(10)   2^Kp    [_  ^^2  +  (p  +  c)2  ~   VoH^ 


;  in  the  limit: 


2ttkp'c 


Art.  260. 


(3)  2xKp'(Vx2  +  a'  -  ■■c). 

(5)  At  the  distance  x  from  the  center,  if  c  is  the  radius  of 
the  circle, 

C/  =  — ^  1^-       o    -^T^ .  where  k  =  3-^— — r  . 

a;  +  c  Jo      VI  —  K-  smV  ^{^c  +  x) 

(6)  C/  =  C2  +  C.  (7)   U  =  mg(zo  -  z). 
(9)    U  —  —    fj{r)dr;  cqiiipotential  surfaces  r  =  c. 

Art.  264. 

(3)  7500,  101.7  X  109.  (5)  150  ft.-lb. 

(4)  18,000  ft.-lb. 

Art.  290. 

(1)  (a)  ilb.;  (6)   11.3  ft. /sec;  (c)  0.63  sec;  {d)  ift.-lb. 

(4)  If  Xq  <  e  nothing  is  changed;  if  Xo  >  e  the  particle 
performs  simple  harmonic  oscillations  about  (^ 

(5)  The  length  I  is  increased  to  I  -\-  e  -{-  Ve(e  +  2/i). 

(7)  42  min.  35  sec. 

Art.  293. 

(2)  The   equation   of   motion   s  =  i*  =  —  g  —  kv'^    gives 
with  A;  =  /^^/jy: 

_  g  iJiVo  cos/Lt^  —  g  sin/^f 
jx/jLio  smiJLt-\-g  cos/zf ' 


ANSWERS  371 

1  ,      g  -\-  kvo'^ 


(4)  ^-i=      ^^ 


^0        -yjg  -\-  kvo^ 

(5)  In  vacuo  v  =  139  ft. /sec,  in  air  v  =  122  ft./sec. 

(6)  s  =  ^  (1  —  e~^0>     ^  =  Voe~''^  =  Vo  —  ks. 

k 


(7)  11  =  ^  (1  -  e-"), 


k^¥  ^^g  -  kv 


Art.  297. 

(2)  The  logarithmic  decrement  is  log  e~^'  =  —  Xf . 

(4)  If  fjL  =^  K,      s  =  Ci  cosk/,  +  Co  siiiKt  +  -^, ,  sin/x^;    if 

K-    —   fJL'^ 

fjL  =  K,      s  —  Ci  cosd  +  Co  sind  +  ^,  sin/c^. 

(5)  The  term  due  to  the  forced  oscillation  is 

a 

a/0c^^^2)2_^  4X2^2  cosm(^  -  h); 

hence  the  oscillation  lags  behind  the  force  by  the  phase 
difference  ixto;  the  amplitude  is  less  than  for  undamped 
oscillations.  The  free  oscillations  (if  any)  will  rapidly  die 
out  so  that  the  motion  soon  approaches  the  state  of  motion 
given  by  the  above  term. 

Art.  302. 

(2)  The  eciuation  of  the  orbit  given  in  Ex.  (1)  is  satisfied 
not  only  by  Xo,  ?/o,  but  also  by  Vi/k,  v^/k;  i.  e.  the  orbit  passes 
not  only  through  the  initial  position  Po,  but  also  through 
the  point  Q(i>i/k,  Vz/k)  which  is  the  extremity  of  the  radius 


372  ANSWERS 

vector  OQ  =  vqIk  parallel  to  fo;  OPq  and  OQ  are  the  con- 
jugate semi-diameters  whose  equations  are  Xoy  =  y^^, 
V\y  =  VoX. 

(4)  The  problem  requires  the  construction  of  the  axes  of 
a  conic  from  a  pair  of  conjugate  diameters. 

(5)  Referring  the  orbit  to  its  axes  we  have  x  =  a  cosd, 
y  =  h  smd  for  the  ellipse  and  x  =  ^0(6"'  +  e"*')  =  a  eoshd, 
y  =  i&Ce"'  —  e""')  =  6  sinh/c^  for  the  hyperbola. 

(6)  From  the  equations  of  Ex.  (5)  it  follows  that  for  the 
ellipse  tan0  =  (6/a)  tauK^  whence  6  =  Kab/r-. 

(8)  Use  the  equations  of  the  conic  in  terms  of  the  eccentric 
angle  (p. 

(9)  (a)  Ellipse;  (6)  hyperbola;  (c)  parabola. 

(10)  The  parabola  x  —  Xo=(vi/v2)iy  —  yo)  —  (2kc/v2^) (?/  —  ?/ o)S 
where  2c  is  the  distance  of  O3  from  the  point  0  that  bisects 
O1O2;  the  midpoint  between  0  and  O3  is  taken  as  origin  and 
OO3  as  axis  of  x. 


(U)  t  =  -tan-i^'^tan^] 


Art.  320. 

(I)  Vo  =   ^fji/ro.  (3)  687  days. 

(4)  As  the  velocity  is  not  changed  instantaneously  we 
have  by  (24),  Art.  314: 

2fjt,      jj.      2iJ.'      ij! 
r.       a        r        a 
whence  a'  is  found. 

(5)  An  ellipse,  with  the  end  of  its  minor  axis  at  the  point 
where  the  change  takes  place. 

(6)  (a)  Ellipse  with  a  =  |r;  (h)  parabola. 

(7)  Differentiate  (24),  Art.  314,  with  respect  to  m  and  a. 

(8)  The  periodic  time  T  would  be  diminished  by  (2/n)r. 

(9)  r  =  Z/(l  +  e  COS0)  gives  x,  y  as  functions  of  9:  hence, 
observing  that  r-^  =  c,  x  =  —  {c/l)  sin^,  y  =  (c'l) (cosd  +  e). 
The  hodograph  is  therefore  the  circle  x^  -\-  {y  —  cejlY 
=  {cjiy,  where  c  ^   -^fxl. 

(10)  1.034  114. 

(II)  t  =   V2aVM(tani0  +  i  tan49). 
(12)  178.73  and  186.52  days. 


ANSWERS  373 


Art.  334. 


(1)  (a)  7h  lb.;  (b)  480  lb.;  (c)  6.4  rev./sec. 

(2)  8i°. 
(4)  32.20. 

(7)    tanS  -  Rco^simpcosip/ig  -  T^co^cos V) ;  44°57'. 
(8)7ilb. 

Art.  339. 

(4)  To  count  the  angles  from  the  highest  point  of  the 
circle  put  t  —  d  =  (p;  then,  putting  h  —  I  =  h',  where  h' 
is  the  height  to  which  the  velocity  at  the  highest  point  is 
due,  we  have  N  =  —  3mg[coS(p  —  f(l  +  h'/l)].  The  par- 
ticle remains  on  the  curve  as  long  as  cos^  >  |(1  +  h' /I) ; 
distinguish  the  cases  h'  =  0. 

(6)  At  the  distance  1.4617a  from  the  lowest  point  of  the 
circle  if  a  is  the  radius. 

Art.  379. 

(c)  tV^;  (d)  tVi^.     (8)  laK 

(9)  ha\ 

(10)  w- 

(11)  ia\  \h\  \c\ 

(12)  i(fli2  +  a,^) 


(2)  (a)  i 

^^:  (6)  W 

(3) 

i/i2. 

(4) 

y>^ 

(5) 

2\a~. 

(6) 

xW- 

(1) 

i.{lv 

'  +  P). 

(2) 

(a)  • 

2^a';  (&)  ■ 

Art.  386. 


(4)  fa^. 
fa^;   (c)  -Jott^.  (5)|(a>  +  a22). 

(3)   (a)  fa^;  (6)  Ja^;  (c)  fa^. 
(6)   (a)  ia^;  (6)  fa^;  (c)  ^^(/i^  +  3a2). 
(8)  ta2. 

(9)  («)     i62.     (5)     1^2.     1(^2   +    ^2). 

(10)  1(6^  +  c2),  i((;2  +  a^),  Ka^  +  ?>^).     (12)  fa^. 

(11)  ia2.  (13)  fa^  +  62. 

Art.  403. 

(1)  The  centroidal  principal  axes  are  perpendicular  to  the 
faces.  The  moments  for  these  axes  are  ^Mih''-  +  c~), 
IMic^  +  a2),     Pf(o2  +  62).         The     central     ellipsoid     is 

(62  +  c2)x2  +  (c2  +  a2)7/2  +  (a2  +  6'-)z2  =  3e4.     For  an  edge 


374  ANSWERS 

2a,  I  =  pf (62  +  c2) ;  for  a  diagonal  /  =  'i-M(b'-c'  +  c"a- 
+  a%^)/{a^  +  6'  +  c2). 

For  the  cube  the  fundamental  ellipsoid  becomes  a  sphere 
of  radius  ^  VOa ;  for  an  edge  of  the  cube,  q"^  =  fa- ;  for  a 
diagonal,  q-  =  fa-. 

(2)  Central  eUipsoid:  (6^  +  c^)^;"  +(c-  +  a-)!]- -\-{a"  +  62)22 
=  Se";  forZ,  g2  =  |(6a2  +  62). 

(3)  Take  the  vertex  as  origin,  the  axis  of  the  cone  as  axis 
of  re;  then  I\  =  y^Ma^;  7i',  t.  e.  the  moment  of  inertia  for 
the  ^2-plane,  =  ^Mh^.  As  for  a  solid  of  revolution  about  the 
axis  oi  X  B'  =  C  and  B  =  C,  we  have  I2  =  I3  =  Hi,  and 
h  =  h  =  //  +  i/i.  Hence,  h  =  h  =  p/(/r  +  ^a^). 
At  the  centroid  the  squares  of  the  principal  radii  are  -y^ci^, 

J?_(4(j2   _[_   /j2\ 

''(4)  A  =  J5  =  C  =  Wa^  D  =  E  =  F  =  IMa";  hence 
momental  ellipsoid:  4(a;2  _|_  ^2  _j_  2,2)  _  3(^2  +  2a;  +  a;|/)  = 
6eVc^^;  squares  of  principal  radii:  ^a^,  \^a'^,  Wa^. 

(5)  g2  =  ia2(i  +  sin^a). 

(6)  /   =  TVpTTflKfa  +  ^  +  2}i^im ;    for  /i  =  a  =  ii7, 

(7)'i' =  /i,  i?  =  /2  +  M.Ti^  C  =  /3  +  M2;i2. 

(8)  The  centroid  may  be  such  xi  point;  if  the  central  ellip- 
soid be  an  oblate  spheroid,  the  two  points  on  the  axis  of 
revolution  at  the  chstance  ±  V(/i  —  l2)/M  from  the  centroid 
are  such  points. 

(9)  The  ellipsoid  must  have  the  same  central  ellipsoid  as 
the  given  body;  its  equation  is  x^/A'  +  if/B'  +  z^/C  =  5/71f , 
where  M  is  the  mass  and  A',  B',  C'  are  the  moments  of  inertia 
for  the  principal  planes  of  the  body  at  the  centroid. 

(10)  p"  =  M/N,  where 

N=^^l^[(q2'+q^'-ql')'+iq^'+q^'-q■y'y'+{qi'+q2'-qz')'f. 

M 

«^  =  f  -77  (<?2-  +  Qs"  -  qi^),  etc. 
P 

Art.  420. 

(2)  §V2a.  (5)  4a. 

(3)  m(|7rr)2.  (6)  yV  lb. 


INDEX. 


(The  numbers  refer  to  the  pages.) 


Absolute  acceleration,  119 

system  of  units,  136 

velocity,  26,  118 

Acceleration,  absolute,  119 

,  angular,  23 

,  center  of,  112 

,  constant,  10-14,  42-43 

directly  proportional  to  dis- 
tance, 18^20 

in  cartesian  coordinates,  38 

in  curvilinear  motion,  35-40 

in  polar  coordinates,  40-42 

in  rectilinear  motion,  7-9 

in  the  rigid  body,  107-116 

inversely      proportional      to 

square  of  distance,  14-18 

of  gravity,  12 

,   normal  or   centripetal,   36, 

109,  111 

,  relative,  119 

,  tangential,  36,  109 

Activity,  212 

Amplitude,  56 

,  correction  for,  54 

Angle  of  friction,  185 

of  repose,  185 

Angular  acceleration,  23 

momentum,    213-216,    270- 

275,  279,  304-306,  313-315 

velocity,  22-24 

,  components  of,  87 

Angular  velocities,  composition  of 
85-91 

,  parallelogram  of,  85 

Anomaly,  eccentric,  240 

,  mean,  242 

,  true,  237,  240 

Aperiodic  motion,  226 

Aphelion,  237 

Areas,  conservation  of,  21G 

,  principle  of,  77 

Arm  of  couple,  159 

Attraction,  187-200 


Beat,  50 
Brachistochrone,  258 


Center,  instantaneous,  99 

of  acceleration,  112 

of  angular  acceleration,   116 

of  force,  72 

of  gravity,  127,  157-158 

of  inertia,  127 

of  mass,  125,  127 

I  ■ of  oscillation,  307 

of  parallel  forces,  156 

of  suspension,  307 

Central  axis,  90,  171 

forces,  229-247 

motion,  72-82 

Centrifugal  force,  251 

Centripetal  acceleration,  109,  111 

■ force,  250 

Centrodes,  99,  100 

Centroid,  125,  158 

Centroidal  line  or  plane,  285 

Circle  of  inflections,  116 

Coefficient  of  friction,  183 

Complanar  forces,  165-169 

Composition  of  angular  velocities, 
85-91 

of  complanar  forces,  165-169 

of  concurrent  forces,  142-145 

of  couples,  159-165 

of  intersecting  rotors,  85-88 

of  parallel  forces,  152-158 

of  parallel  rotors,  88-91 

of  simple  harmonic  motions, 

59-63 

of  velocit  ies,  27-29 

Compound  harmonic  mot  ion, 59-63 

harmonic  wave  motion,  66 

pendulum,  306 

Concurrent  forces,  142-145,  149- 
151 

Cone,  cquimomental,  292 

Cone  of  friction,  185 

Confocal  conies,  298-299 

quadric  surfaces,  299-301 

Conservation  of  angular  momen- 
tum or  areas,  216,  275,  348 

of  linear  momentum,  273, 

348 


375 


376 


INDEX 


Conservation  of  energy,  210,  212, 

227,  347 
Conservative  forces,  196,  210,  355 
Constant  acceleration,  10-14,  42- 

43 

of  gravitation,  187-188,  235 

Constrained  motion  of  a  particle, 

248-267 

motion  of  a  system,  348-360 

Constraints,     178-182,     260-267, 

348-360 
Coriolis,  theorem  of,  119,  335 
Couples,  151,  158-165 
Cross-product,  85 

D'Alembert's  principle,  259,  347- 

349 
Damped  oscillations,  224-227 
Damping  ratio,  228 
Decrement,  logarithmic,  228 
Degrees  of  freedom,  179,  268-269, 

352-354 
Density,  121-123 
Derived  units,  121 
Deviation,  moment  of,  281 
Dimensions  of  acceleration,  9 

■ of  force,  135 

of  momentum,  133 

of  power,  212 

of  velocity,  5 

of  work,  201 

Dot-product,  110 
Dynamics,  1,  120 
Dyne,  135 

Eccentric  anomalj^,  240 

Effective  force,  259 

Efficiency  of  a  machine,  212 

Elevation,  angle  of,  44 

Ellipsoid,  central  and  fundamen- 
tal, 295 

,  momental,  290 

of  gyration,  295 

of  inertia,  290,  295 

,  reciprocal,  295 

Elliptic  harmonic  motion,  69 

Energy,  kinetic,  137 

,  potential,  210 

,  total,  210 

Epoch-angle,  56 

Equation  of  the  center,  243 

Equations  of  linear  and  angular 
momentum,  269-271 


Equilibrium,  144 

of  complanar  forces,  165 

of  concurrent  forces,  144-146 

of  general  system  of  forces, 

170,  176 

of  parallel  forces,  156-157 

Equimomental  cone,  292 

Equipotential  surfaces,  199 

Equivalent  simple  pendulum,  307 

Erg,  202 

Euler's  angles,  318 

equations    of    motion,    316- 

317,  323 

Focal  ellipse  and  hyperbola,  300 
Foot-pound,  foot-poundal,  202 
Force,  133  _ 
Force-fvmction,  196 
Force-polj'gon,  144,  154 
Forced  oscillations,  227-229 
Forces,  central,  229-247 
,  centrifugal  and  centripetal, 

250-251 

,  complanar,  165-169 

,  conservative,   196,  210,  355 

,  effective,  259 

,  general  system  of,  170-178 

,  parallel,  151-158 

Free  oscillations,  217-221 
Freedom,  degrees  of,  179,  268-269, 

352-354 
Frequency,  58 
Friction,  182-186 
Friction  angle,  185 

cone,  185 

Fundamental  units,  121 
Funicular  polygon,  155 

Generalized     coordinates,     263, 
352 

Gradient,  199 

Gravitation,  constant  of,  187-188, 
235 

,  law  of,  187  _ 

system  of  units,  136 

Gyration,  radius  of,  281 

Hamilton's  principle,  359-360 
Harmonic  motion,  16-18,  52-70 
Head  or  height  due  to  velocity,  12 
Heavy  symmetric  top,  327 
Hcrpolhode,  321 


INDEX 


377 


Heterogeneous  mass,  121 
Hodograph,  37 
Holonomic,  349 
Homogeneous  mass,  121 
Hooke's  law  of  elastic  stress,  218 
Horse-power,  213 

Impulse,  134,  275,  278-279 
Indeterminate    multipliers,    259- 

263,  351 
Inertia,  133 

,  ellipsoid  of,  290,  295 

,  moment  of,  280 

,  product  of,  281 

,  radius  of,  281 

,  spherical  points  of,  297 

Instantaneous  axis,  84 

center,  99 

Invariable  direction  and  plane,  275 
Invariant  of  system  of  forces,  173 
Isochronous,  50 

Kepler's  equation,  242 

laws,  75,  78,  79 

Kinematics,  1,  3-119 
Kinetic  energy,  137 

friction,  183 

potential,  356 

Kinetics,  1,  207-360 
Kinetostatics,  250 

Lagrange's  equations  of  motion, 

263,  352 
Lagrangian  coordinates,  263,  353 

function,  356 

Laplace's  equation,  198 

invariable  plane,  275 

Linear  density,  123 

mass,  122 

momentum,  273,  348 

— —  velocity,  24,  83,  87 
Lines  of  force,  199 
Lissajous's  curves,  70-72 
Logarithmic  decrement,  228 

Mass,  120-123 

moment,  124 

Mean  anomaly,  242 

motion,  240 

Mechanics,  1 

Method   of   indeterminate  multi- 
pliers, 259-203,  351 


Moment  of  a  couple,  159,  162 

■ of  a  force  about  an  axis,  177 

of  a  force  about  a  point,  150 

of  inertia,  280. 

of  mass,  124 

of  momentum,  213 

Momental  ellipsoid,  290 
Momentum,  132 

,  angular,   213-216,   270-275, 

279,  304-306,  313-315 

,  linear,  277,  348 

Motion,  mean,  240 

Newton's  laws  of  motion,  139-141 

law  of  universal  gravitation, 

187 
Normal  acceleration,  36,  109 

Oscillations,  damped,  224-227 

,  forced,  227-229 

,  free,  217-221 

Parallel  forces,  151-158 

Parallelogram  of  angular  veloc- 
ities, 85 

of  forces,  143 

of  linear  velocities,  28 

Particle,  123 

Pendulum,  compound,  306 

,  simple,  47-54,  253 

PerUielion,  237 

Period,  56 

Periodic  time,  79,  240 

Permanent  axes  of  rotation,  311 

Phase,  phase-angle,  56 

Planetary  motion,  80-82,  233-247 

Polar  reciprocal  of  momental 
ellipsoid,  294-295 

Polhode,  321 

Potential,  196-200 

energy,  210 

Poundal,  136 

Power,  212 

Precession,  326 

Principal  axes,  290,  311 

■ ,  distribution  of,  297-303 

Principle,  d'Alembert's,  259,  347- 
349 

■ ,  Hamilton's,  359-360 

of    angular    momentum    or 

of  areas,  77,  213 


378 


INDEX 


Principle  of  conservation  of  angu- 
lar momentum  or  of  areas, 2 16, 
275,  348 

of    conservation    of    energy, 

210,  212,  227,  347 

of     conservation     of     linear 

momentum,  273,  348 

of  independence  of  transla- 
tion and  rotation,  275 

of  kinetic  energy  and  work, 

77,  208 

of  virtual  velocities,  206 

of  virtual  work,  205 

Problem  of  two  bodies,  244-247 

Products  of  inertia,  281 

Quantity  of  motion,  132 

Radius  of  inertia  or  of  gyration, 

281 
Range  of  projectile,  45 
Reactions,  179,  249,  308-312 
Reciprocal  ellipsoid,  295 
Relative  acceleration,  118,  335 

motion,  117-119,  335-345 

velocity,  26,  117     , 

Resistance  of  a  medium,  221-224 
Resultant  force,  142 

velocity,  27 

Rigid  body,  21,  149 
Rotation,  21,  84 
Rotor,  84,  139 
Rotor-couple,  90 

Scientific  system  of  units,  136 
Sci'ew  motion,  91 
Seconds  pendulum,  51 
Sectorial  velocity,  33,_  73 
Simple  harmonic  motion,  54-58 

harmonic  wave  motion,  65 

mathematical  pendulum,  47- 

54,  253-255 
Specific  density,  specific  gravity, 

122 
Static  friction,  183 
Statics,  1,  120-206 
Surface  density,  123 

mass,  122 

Swing,  50 

Tangential  acceleration,  36,  109 
Tautochrone,  255-258 


Theorem  of  Coriolis,  119,  335 

of  moments,  150,  153 

of  Varignon,  150 

Time  of  flight  of  projectile,  46 
Top,  327 
Torque,  159 
Total  energy,  210 

reaction,  184 

Translation,  21,  S3,  108 

Triangle  of  forces,  143 

True  anomaly,  237,  240 

Twist,  91 

Two  bodies,  problem  of,  244-247 

Uniformly  accelerated  motion,  10 

-14,  23-25 
Unit  of  acceleration,  9 

of  density,  122 

of  force,  135 

of  mass,  121 

of  momentum,  133 

of  power,  212 

of  velocity,  5 

of  work,  202 

Units,  fundamental  and  derived, 

121 

,  systems  of,  136 

Universal  gravitation,  187 

Varignon's  theorem,  150 
Vector,  26 
Velocity,  3 

,  absolute,  26,  118 

,  angular,  22-24,  84 

,  body-,  117 

,  linear,  24,  83,  87 

— ^-  of  propagation  of  wave,  64 

of  rotation,  84 

of  translation,  83 

,  relative,  26,  117 

,  sectorial,  33,  73 

Virtual  displacement,  203 

velocities,  principle  of,  206 

work,  201,  203 

work,  principle  of,  205 

Wave  length,  64 

motion,  63-67 

Weight,  157-158 

Work,  201 

,  virtual,  203 


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The  Elementary  Part  of  a    Treatise  on  the 
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Being  Part  I.  of  a  Treatise  on  the  Whole  Subject. 

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Cloth,  diagrams,  12mo,  ^2.00  net. 

"A  Text-book  for  Engineering  Students,  which  aims  to  give 
abundant  application  to  every  principle  developed,  and  in  this 
application  to  use  problems  such  as  an  engineer  would  be 
likely  to  encounter.  In  this  endeavor  he  has  been  success- 
fill."  — Engineering  News. 

The  Principles  of  Mechanics 

By  FREDERICK  SLATE,   Professor  of  Physics  in  the 

University  of  California. 

Cloth,  12mo,  $1.90  net. 

An  elementary  exposition  for  students  of  physics,  the  material 
of  which  has  been  shaped  by  the  influence  of  three  desires  : 
first,  to  select  the  subject-matter  with  close  reference  to  the 
needs  of  college  students  ;  second,  to  bring  the  instruction  into 
adjustment  with  the  actual  stage  of  their  training  ;  and,  third, 
to  aim  continually  at  treating  mechanics  as  a  system  of 
organized  thought,  having  a  clearly  recognizable  culture 
value. 


An  Elementary  Treatise  on  the  Mechanics  of 
Machinery  with  Special  Reference  to  the 
Mechanics  of  the  Steam-Engine 

By  JOSEPH  N.  LECONTE,  Instructor  in  Mechanical 
Engineering,  University  of  California,  Associate  Member 
of  the  American  Institute  of  Electrical  Engineering. 

Cloth,  12mo,  $2.25  net. 


The  Advanced  Part   of   a    Treatise  on  the 
Dynamics  of  a  System  of  Rigid  Bodies 

Being  Part  II.  of  a  Treatise  on  the  Whole  Subject. 

Sixth  Edition,  revised  and  enlarged. 

Cloth,  8vo,  with  diagrams,  etc. ,  $3. 75  net. 

The  general  plan  of  the  book  is  the  same  as  before,  with  the  several  chapters 
as  independent  as  possible  to  enable  the  reader  to  choose  his  own 
order  of  study.  The  student  will  see  by  a  glance  at  the  table  of  con- 
tents how  many  and  how  various  are  the  applications  of  dynamics 
and  it  is  not  to  be  supposed  that  this  list  is  exhaustive,  so  rapidly 
does  the  subject  grow. 

A  Treatise  on  Dynamics 

With  Examples  and  Exercises 

By  ANDREW  GRAY,  LL.D.,  F.R.S.,  Professor  of 
Natural  Philosophy  ni  the  University  of  Glasgow,  and 
JAMES  GORDON  GRAY,  D.Sc.,  Lecturer  on  Physics 
m  the  University  of  Glasgow. 

Cloth,  12mo,  $8.25  net. 

The  Elements  of  Graphic  Statics 

By  L.  M.  HOSKINS,  Professor  of  Applied  Mechanics  in 
the  Leland  Stanford  Junior  University.     Revised  edition. 
Cloth,  8vo,  diagrams,  $2.25  net. 
A  Text-book  for  Students  of  Engineering. 


Published  by 

THE  MACMILLAN  COMPANY 

Sixty-four  and  Sixty-six  Fifth  Ave.  NEW  YORK 


Elements  of 
Theoretical    Mechanics 

By   ALEXANDER    ZIWET 

Professor  of  Alathemafics  in  the  University  of  Michigan 
Revised  Edition 

Especially  Designed  for  Students  of  Engineering 


Cloth,  8i'u,  $^.00  net 

This  revised  edition  in  one  volume  represents  essentially  the 
required  course  in  theoretical  mechanics  as  given  in  the 
Engineering  Department  of  the  University  of  Michigan.  In 
order  to  keep  within  the  bounds  of  a  thi'ee-hour  course  ex- 
tending through  a  year  and  within  the  reach  of  the  mathe- 
matical attainments  of  a  second  or  third  year' s  college  student 
it  seemed  best  to  confine  the  treatment  largely  to  problems 
in  one  or  two  dimensions  (except  in  statics).  Thus  the 
motion  of  a  rigid  body  about  a  fixed  point  had  to  be  omitted, 
in  spite  of  its  importance.  But  rectilinear  motion  and  rota- 
tion about  a  fixed  axis  have  received  more  ample  treatment, 
and  at  least  some  illustrations  of  plane  motion  have  been  given. 

"I  can  state  without  hesitation  or  qualification  that  the  work  is 
one  that  is  unexcelled,  and  in  every  way  surpasses  as  a  text-book 
for  class  use  all  other  works  on  this  subject ;  and  moreover,  I  find 
the  students  all  givina:  it  the  highest  praise  for  the  clear  and  in- 
teresting manner  in  which  the  subject  is  treated." 

M.  J.  McCuK,  M.S.,  C.E.,  University  of  Notre  Dame,  Ind. 


Published  by 

THE  MACMILLAN  COMPANY 

Sixty-tour  and  Sixty-six  Fifth  Ave.  NEW  YORK 


UC  SOUTHERN  REGIONAL  LIBRARY  FAClUn 


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